Extending the isolated horizon phase space to string-inspired

gravity models

by

Tomas Liko

A thesis

presented to Memorial University of Newfoundland

in fulfillment of the

thesis requirement for the degree of

Doctor of P hilosophy

in

Theoretical Physics

St. John's, Newfoundland, Canada, 2008

© Tomas Liko 2008

ABSTRACT

An isolated horizon (IH) is a null hypersurface at which the geometry is held fixed. This

generalizes the notion of an event horizon so that the black hole is an object that is in local

equilibrium with its (possibly) dynamic environment. The first law of IH mechanics that

arises from the framework relates quantities that are all defined at the horizon.

IHs have been extensively studied in Einstein gravity with various matter couplings and

rotation, and in asymptotically flat and asymptotically anti-de Sitter (ADS) spacetimes in

all dimensions D 2:: 3. Motivated by the nonuniqueness of black holes in higher dimensions

and by the black-hole/string correspondence principle, we devote this thesis to the extension

of the framework to include IHs in string-inspired gravity models, specifically to Einstein­

Maxwell-Chern-Simons (EM-CS) theory and to Einstein-Gauss-Bonnet (EGB) theory in

higher dimensions. The focus is on determining the generic features of black holes that are

solutions to the field equations of the theories under consideration. To this end, we construct

a covariant phase space for both theories; this allows us to prove that the corresponding

weakly IHs (WIHs) satisfy the zeroth and first laws of black-hole mechanics.

For EM-CS theory, we find that in the limit when the surface gravity of the horizon goes

to zero there is a topological constraint. Specifically, the integral of the scalar curvature

of the cross sections of the horizon has to be positive when the dominant energy condition

ii

is satisfied and the cosmological constant A is zero or positive. There is no constraint

on the topology of the horizon cross sections when A < 0. These results on topology of

IRs are independent of the material content of the stress-energy tensor, and therefore the

conclusions for EM-CS theory carry over to theories with arbitrary matter fields (minimally)

coupled to Einstein gravity.

In addition, we consider rotating IRs in asymptotically ADS and flat spacetimes, and

find the restrictions that are imposed on them if one assumes they are supersymmetric. For

the existence of a null Killing spinor in four-dimensional N = 2 gauged supergravity we show

that ADS supersymmetric isolated horizons (SIRs) are necessarily extremal, that rotating

SIRs must have non-trivial electromagnetic fields, and that non-rotating SIRs necessarily

have constant curvature horizon cross sections and a magnetic (though not electric) charge.

When the cosmological constant is zero then the gravitational angular momentum vanishes

identically and the corresponding SIRs are strictly non-rotating. Likewise for the existence

of a null Killing spinor in five-dimensional N = 1 supergravity, we show that SIRs (in

asymptotically flat spacetimes) are strictly non-rotating and extremal.

For EGB theory we restrict our study to non-rotating WIRs and show explicitly that the

expression for the entropy appearing in the first law is in agreement with those predicted

by the Euclidean and Noether charge methods. By carefully examining a concrete example

of two Schwarzschild black holes in a flat four-dimensional spacetime that are merging, we

find that the area-increase law can be violated for certain values of the GB parameter. This

provides a constraint on the free parameter.

iii

ACKNOWLEDGEMENTS

I would like to thank my Ph.D advisor Ivan Booth for collaborations, discussions and most

importantly for supporting my ideas. I would also like to thank Abhay Ashtekar, Dumitru

Astefanesei, Kirill Krasnov and Don Witt for correspondence related to some of the work

presented in this thesis, and especially Kirill for taking time to answer my naive questions

about quantum gravity.

This work was funded by the Natural Sciences and Engineering Research Council of

Canada (NSERC) and by funds from the School of Graduate Studies (SGS). Participation

at conferences was supported in part by NSERC and by travel funds received from SGS

and the Department of Physics. Participation at the Topical Workshop on Black Holes and

Fundamental Theory at Penn State University was funded by NSERC.

The format of this thesis is based on the style files that were previously used by Sanjeev

Seahra for his thesis at the University of Waterloo.

My deepest gratitude goes to my daughters Kristijana and Veronika for always reminding

me that there is more to life than physics, to my wife Jana for keeping me focused on getting

my work done quickly, and to my parents Maria and Stefan for continued encouragement.

iv

CONTENTS

Abstract

Acknowledgements

List of Figures

Notation and Conventions

Authorship Statement

1 Introduction

1.1

1.2

1.3

Statement of the problem

Motivation for thesis research

Overview and main results . .

2 Isolated Horizons in EM-CS Theory

2.1

2.2

First-order action for EM-CS theory

Boundary conditions . . . . . .

2.3 Variation of the boundary term

2.4 Covariant phase space . . . . .

v

ii

iv

viii

ix

XV

1

1

5

16

16

19

22

25

2.5 Angular momentum and the first law .

2.6 Rotation in ADS spacetime . .. . . .

2. 7 A topological constraint from extremali ty

3 Supersymmetric isolated horizons

3.1

3.2

3.3

Black holes and Killing spinors .

Killing spinors in four dimensions .

Killing spinors in five dimensions

3.4 Interpretation . . . . . . . . . . .

3.5 Relation to asymptotically flat solutions

Appendix: an alternate formalism in four dimensions .

4 Isolated Horizons in EGB Theory

4.1

4.2

4.3

First-order action for EGB theory

Modified boundary conditions and the zeroth law

Variation of the boundary term . . . . .

4.4 Covariant phase space and the first law

4.5 Comparison with Euclidean and Noether charge methods

4.6 Decrease of black-hole entropy in EGB theory . . . . . . .

5 Prospects

5.1 Summary

5.2 Classical applications to EM-CS and EGB theories

A Black Hole M echanics: An Overview

vi

26

29

33

38

38

39

44

47

51

55

57

57

58

60

64

66

68

71

71

72

76

A.l Thermodynamics . .. . .. . . ... . . .

A.2 Black-hole mechanics: global equilibrium.

B Differential forms

B.l Definitions .. .

Bibliography

Vll

76

78

82

82

99

LIST OF FIGURES

2.1 The spacetime manifold M and its boundaries. . . . . . . . . . . . . . . . . 19

viii

NOTATION AND CONVENTIONS

In this thesis, the geometrical setting is a D-dimensional Lorentzian spacetime manifold M

bounded by two spacelike partial Cauchy surfaces, M- and M+, which are asymptotically

related by a time translation and extend from the internal boundary 6. (with 6. nM ~ §D- 2

for some compact (D- 2)-dimensional space sD-2) to the boundary at infinity ~. Below

is a table of symbols that are used in this thesis.

Symbol

a, b, . .. E {0, ... , D - 1}

I, J, . .. E {0, . . . , D - 1}

i,j, ... E {2, ... ,D- 1}

L E {1 , ... , l(D- l)/2J}

Description

Timelike conformal boundary of asymptotically anti-de Sitter

spacetimes

Conformal completion of asymptotically anti-de Sitter spacetimes

Compact (D - 2)-dimensional cross section of .Jf such that .Jf n

M ~ cv-2

Spacetime indices on M

Internal Lorentz indices in the tangent space of M

Spatial indices on sD- 2

Rotation index; corresponds to l ( D - 1) / 2 J independent rotation

parameters in D dimensions (with l·J denoting "integer value

of")

ix

Symbol

A,B, ... E{1,2}

A',B', . .. E {1,2}

a,{J, ... E {1, 2}

9ab

Rabcd

Rab = Rc acb> R = gab Rab

Gab = Rab - (R/2)9ab: Tab

Aa, Fab = 8aAb - 8bAa

9ab

Cabcd

Eab

n

A,L

c, li, kB , Go

A

T/IJ = diag(-1, 1, .. . , 1)

Description

Spinor indices labelling components of two component spinors in

four dimensions

Dual spinor indices labelling components of two component

spinors in four dimensions

Spinor indices labelling ~ymplectic spinors in configuration space

of spinors in five dimensions

D-dimensional metric tensor on M ; sig. (- + · · · +)

Riemann curvature tensor determined by 9abi employing the con­

vention of Wald (1984)

Ricci tensor and Ricci scalar

Einstein tensor and stress-energy tensor

Electromagnetic vector potential and Faraday field tensor

D-dimensional metric tensor on M

Weyl tensor of 9ab

Electric part of Cabcd

Conformal factor relating 9ab and 9ab

Cosmological constant and anti-de Sitter radius

Physical constants: speed of light, Planck constant, Boltzmann

constant, gravitational constant

Coupling constants: ko = 81rG o , Gauss-Bonnet

Chern-Simons parameter; 1 if D is odd and 0 if D is even

Internal Minkowski metric in the tangent space of M

Coframe; a set of D orthogonal vectors defined by the condition

9ab = TJI Je/ ® eb J

X

Symbol

19 (i)

Qab = 9ab + fanb

Qab = 9ab + fanb + ebna

Wa, A, K.(t) = eawa. 'l>(t)

eaAa

f.J, ... Io

f = 19(1) (\ ... /\ 19( D - 2)

K,K'

Description

Null normal to ~; f' "' e if f ' = zf, z constant

Auxiliary null normal to ~ normalized such that naea = -1

(D - 2) spacelike vectors satisfying e · 19(i) = n · 19(i) = 0 and

normalized such that 19(i) ·19(j) = Oij

(D- I )-dimensional degenerate metric on ~; sig. (0 + ... +)

Induced metric on §D-2

Induced normal connection, area, surface gravity and electromag­

netic scalar potential on ~

Denotes pullback to ~

Denotes equality restricted to ~

Gravitational SO(D - 1, 1) connection

Gravitational curvature two-form defined by A 1 J

(D - m)-form determined by the coframe

Spacetime volume element

Totally antisymmetric Levi-Civita tensor

Area element of §D- 2

Timelike Killing fields in globally stationary asymptotically anti­

de Sitter spacetimes

Killing vector field that generates a symmetry (i .e. time transla­

tion etc) in asymptotically anti-de Sitter spacetime

Unit timelike normal to cD- 2

Area form on cD-2

Gradient of n on J

xi

Symbol

c

f,g, V" , W", wab

:F, vr, iplJ

Ej_, Bj_

X" E T*(§v)

Description

Electromagnetic (U(1)) connection one-form and associated cur­

vature two-form

Electric and magnetic charges

Anticommuting Dirac spinor in four dimensions

Commuting symplectic Majorana spinors in five dimensions

Spinor structure associated withE"'; norm. € 12 = €12 = 1

Gamma matrices in four dimensions

Gamma matrices in five dimensions

Totally antisymmetric product of D gamma matrices

Hermitian conjugate t and transpose T of the matrix M

Charge congujation matrix

Bilinear covariants defined by products of € and 1"

Bilinear covariants defined by products of € 0 and f 1

Electric and Magnetic fluxes through b.

Function describing fiO\vs of electromagnetic radiation along b. ~

Rx §v

Killing spinor in four dimensions

Spinor structure associated with '!f;AA'; norm. €12 = €12 = 1

Maxwell spinor and its complex congujate

Killing vector which is null on b.

Complex-valued function defined by j3A = JCG.A

Complex-valued function defined by ¢AB = ¢1Y.AIY.B

Generic field variables: tensor field, differential form

xii

Symbol

£eiii = f.JdiiJ + d(f...Jiii)

ow,o~

E[·]

J[· ,·]

0 [·,·]

¢~

~a= zea +I:. n.¢~

S, Q, :J.

Rabcd

.fli,K,

(a = ta +I:. n.m~

Description

Derivative operators: partial derivative, covariant derivative,

gauge covariant derivative, intrinsic covariant derivative on 6.

and exterior derivative

Contraction of a vector with a differential form

Hodge dual of ~

Lie derivative

first variation; also used to denote small changes

Equations of motion

Surface terms

Symplectic structure

L(D - l)/2J angular velocities of 6.

Globally defined rotational spacelike Killing fields on M

Restrictions of <p~ to 6.

Evolution vector field; spacelike on 6.

Conserved charges on 6.: entropy, electric charge and angular

momenta

Riemann curvature tensor associated with the metric Qab

Scalar curvature of § 0 - 2

Euler characteristic of § 2 with genus g

Area and surface gravity of a stationary black hole

Evolution vector field for stationary spacetimes with ta a timelike

Kill ing vector and m~ globally defined rotational spacelike Killing

vectors; the hypersurface at which (a(a = 0 defines the Killing

horizon with angular velocities n.

Xlll

Symbol Description

<E , .Q , J . Conserved charges of a stationary black hole (measured at infin­

ity): energy, electric charge and angular momenta

V(k)N - I = 7rN/2 jf(N/2+1) Volume of an (N - I)-dimensional space SN-I = 8~)1 of con­

stant curvature

AN- I = 21fN/2 / f (N/2)

Y, T

Metric on §N - I

curvature index of §N-I: k = 1 corresponds to positive constant

curvature, k = - 1 corresponds to negative constant curvature,

and k = 0 corresponds to zero curvature

radius of event horizon

Surface area of a unit (N- I)-sphere; f (N /2) gamma function

Thermodynamic parameters: entropy and temperature

XIV

AUTHORSHIP STATEMENT

This thesis is based on the following four articles:

• Liko T and Booth I 2007 Isolated horizons in higher-dimensional Einstein-Gauss­

Bonnet gravity Class. Quantum Grav. 24 3769

• Liko T 2007 Topological deformation of isolated horizons Phys. Rev. D 77 064004

• Liko T and Booth I 2008 Supersymmetric isolated horizons Class. Quantum Grav.

25 105020

• Booth I and Liko T 2008 Supersymmetric isolated horizons in ADS spacetime Phys.

Lett. B 670 61

I am first author of the first three articles. I wrote the manuscripts and Ivan helped with

the editing. I submitted and revised these articles to address the issues that were raised

by the referees. I initiated the projects and did most of the calculations. The extremality

condition (2.54) is due to previous work by Ivan and his collaborator Stephen Fairhurst.

The identity (4.18) for the Riemann tensor was proved by Ivan. The latest article was a

joint collaboration between Ivan and myself. I initiated the project, but the calculations

and preparation of the manuscript were done together.

XV

c:::-1 Introduction

"What is mind? No matter. What is matter? Never mind. " ,...., H J Simpson

1.1 Statement of the problem

It has been appreciated for some time that a black hole behaves as a thermal object and

has a macroscopic entropy YsH, the Bekenstein-Hawking entropy, that is proportional to

the surface area .sd of the event horizon (Bekenstein 1973; Bekenstein 1974; Hawking 1975).

This fact is a very beautiful example of the profound relationship between the classical and

quantum aspects of the gravitational field , and is one of the main reasons why the study

of black holes continues to be one of the most interesting areas of research in gravitational

theory. It is also the main reason to believe that the gravitational field should have a

quantum description. One of the goals of all the different approaches to quantum gravity is

to identify the microscopic degrees of freedom that account for the entropy, and to obtain the

area-entropy relation from first principles using statistical mechanics. If it turned out that

gravity cannot be quantized, then this fact would provide a very striking counterexample

1

Chapter 1. Introduction 2

to our belief that thermal properties of any object are described quantum mechanically in

terms of the microstates of the corresponding system.

To get a general feeling for the problem, it is worth looking at the entropy with a

concrete example. First, let us write down the expression with all physical constants. For

a black hole in Einstein gravity this is

(1.1)

with kB the Boltzmann constant and l~ the Planck "area" defined by the speed of light c,

D-dimensional gravitational constant G D and Planck constant li. Let us further consider a

Schwarzschild black hole of one solar mass M0 = 1.989 x 1030 kg in four dimensions. The

spacetime for this solution in spherical coordinates is the line element

The event horizon has radius r = 2G4M 0 jc? and the surface area is then .fll = 321l'G~M~/c4 .

This gives a numerical value of

(1.3)

for the entropy of the black hole. The number of quantum states N that this entropy

corresponds to is therefore

N = exp (~) = exp(2.098 x 1077), (1.4)

which is a huge number by any standards. For comparison, we note that the number .Y / ks

is on the same order of magnitude as the estimated total number of nucleons in the universe!

The problem is to answer the following question : What are the microscopic degrees of

freedom that account for the entropy of the black hole'? The Schwarzschild solution is static,

Chapter 1. Introduction 3

which implies that the degrees of freedom cannot be gravitons. They must be described by

nonperturbative configurations of the gravitational field .

The leading approaches to quantum gravity that have been most successfully applied

to the problem of black-hole microstates are loop quantum gravity (LQG) (Ashtekar and

Lewandowski 2004) and superstring theory (ST) (Aharoney et al 2000).

• Loop quantum gravity. Here one counts the states arising from punctures where spin

networks traversally intersect a surface that is specified in the quantized phase space

with a set of boundary conditions (Ashtekar et al 1998; Ashtekar et al 2000a). This

surface represents the black hole horizon and is intrinsically flat. Curvature is indue d

at the punctures where the spin networks intersect the surface and give it "quanta of

area". The LQG framework has been successful in describing the statistical mechanics

of all black holes (with simple topologies) in four dimensions, but up to a single free

parameter that enters into the classical phase space as an ambiguity in the choice of

real self-dual connection (Barbero 1996; Immirzi 1997; Rovelli and Thiemann 1997).

In order for the framework to produce the correct coefficient that matches the one for

.Yin (1.1) , this parameter must be fixed to a specific value which depends on how the

state-counting is done. See e.g. (Tamaki and Nomura 2005) . It was recently pointed

out, however, that if the ewton constant as well as the surface area of the black hole

are renormalized then the entropy may be the same for all values of the parameter

(Jacobson 2007) . Also, certain special properties of four-dimensional spacetimes have

to be exploited within the framework, which are crucial for the calculations to work

at all. This makes it difficult to extend the framework to higher dimensions.

Chapter 1. Introduction 4

• Superstring theory. There are two (independent) approaches to the problem here.

The first is the D-brane picture (Maldacena 1996; and references therein), whereby

one counts the states of a particular quantum field theory on a configuration of D­

branes which forms a black hole in the limit when the string coupling is increased.

The second is the anti-de Sitter/conformal field theory (ADS/CFT) picture (Witten

1998a; 1998b), whereby a black hole in a five-dimensional ADS spacetime is described

by a conformally invariant SU(N) super Yang-Mills theory; here the states accounting

for the entropy are the quantum states of the CFT. Both of these approaches have

been successful in describing the statistical mechanics, with the exact coefficient for

the area-entropy relation, but for a very limited class of black holes: extremal and

near-extremal in the D-brane picture while very small black holes (corresponding to

high temperature limit) in the ADS/CFT picture. In particular, astrophysical black

holes such as those described by the solution (1.2) are not among the class of black

holes that are described in the ST approaches.

The LQG and ST approaches are very different, both philosophically and in the methods

that are used for quantization. LQG on the one hand is a background independent canon­

ically quantized theory of pure gravity in four dimensions, while ST on the other hand is

a quantum field theory over a fixed nondynamical background in higher dimensions that is

supposed to describe all interactions as well as gravity. It is unclear, and surprising, that

such different approaches all lead to the same answer. This is an instance of the "problem

of universality" which has been advocated for some t ime now by Carlip (2007). Essentially,

the entropy of a black hole may be fixed universally by the diffeomorphism invariance of

Chapter 1. Introduction 5

general relativity.

1.2 Motivation for thesis research

The fact that the ST approaches give the exact coefficient for the entropy of a black hole is

truly remarkable, despite that they do so for such a limited clas of solutions. Nevertheless,

the ST approaches are the most favored because they explain the entropy dynamically,

and also from an aesthetically pleasing point of view that ST is a unified theory of all

interactions. Despite all successes though, a number of problems remain. Among the more

serious ones are black hole nonuniqueness in higher dimensions and an inconsistency that

has been overlooked in the black-hole/string correspondence principle.

• Black-hole nonuniqueness. In a four-dimensional asymptotically flat spacetime, a

charged and rotating black hole is uniquely described by its conserved charges; the

only unique solution is the Kerr-Newman metric (Robinson 1973). This is a statement

of the black-hole uniqueness theorem, and is a striking property of the simplicity of

black holes in nature. The advent of ST revolutionized our view of the universe,

for example with the requirement of extra spatial dimensions. For a long time it was

generally assumed that the properties of four-dimensional black holes, particularly the

uniqueness theorem, simply carry over to higher dimensions as well. The black ring

solution (Emparan and Reali 2002) that describes a rotating black hole with horizon

topology 8 1 X S2 in five dimensions, was the first counter-example to the uniqueness of

black holes in asymptotically flat spacetimes. Specifically, the uniqueness theorem fails

(in five dimensions) because the conserved charges of the ring can coincide with the

Chapter 1. Introdu·:.:tion 6

conserved charges of a rotating black hole with horizon topology S3 (Myers and Perry

1986). The natural question that should be investigated is therefore the following:

What properties of black holes in four dimensions carry over to higher-dimensional

spacetimes? More specifically, we should ask the following question: What are the

generic features of black holes in higher-dimensional spacetimes in general and within

the ST framework in particular? An ideal method of investigating such questions is

to employ a covariant phase space of all solutions to the equations of motion for a

given action principle.

• Black-hole/string correspondence principle. The methods that are employed in ST

lead to a first law of black-hole mechanics that relates quantities at the event horizon

and quantities ·defined at infinity. This "hybrid" relation appears to be unphysical in

ST from the point of view of the black-hole/string correspondence principle (Susskind

1993) , which states that there is a smooth transition from a black hole to a string

in the limit when the string coupling is decreased. For this correspondence principle

to work, the entropies of the black hole and string are required to be equal for a

particular value of the string coupling constant because the entropy of the black hole

is proportional to the mass squared and the entropy of the string is proportional to

the mass (Horowitz and Polchinski 1997). However, while the mass of the black hole

is measured at infinity, the mass of the string is determined by the string coupling and

tension which are intrinsic quantities of the string state in the sense that no reference

needs to be made to infinity at all. Therefore the conserved charges of the black hole

state should not be defined at infinity.

Chapter 1. Introduction 7

The moral to be extracted from the above considerations is that a framework for black

holes in ST should be employed that is both quasilocal and general enough to allow for a

large class of solutions to be investigated. Remarkably, such a framework doe exist! This

is the isolated horizon (IH) framework (Ashtekar and Krishnan 2004). The classical theory

of IHs was motivated by earlier considerations by Hayward (1994), but the framework

is considerably different as covariant phase space methods (Witten 1986; Crnkovic 1987;

Crnkovic and Witten 1987; Crnkovic 1988; Lee and Waltl 1990; Ashtekar et al 1991; Wald

and Zoupas 2000) are employed in th former case. All the quantities that appear in the

first law of IH mechanics are defined intrinsically at the horizon. The concept of such a

surface generalizes the notion of a Killing horizon in stationary spacetim to much more

general and therefore physical spacetimes that may include exterual radiation fields that

are dynamical. Examples of such system in general relativity are given by the so-called

Robinson-Trautman spacetimes (A htekar et al 1999; Lewandowski 2000).

The IH framework may fit naturally into ST and the black-hole/string correspondence

principle. The work presented here is a first step towards extending the IH phase space

beyond Einstein gravity so that a quasilocal description of black holes may be realized

within the context of ST. In this thesis the framework is extended first to Einstein-Maxwell-

Chern-Simons (EM-CS) theory (Booth and Liko 2008; Liko and Booth 2008) and then to

Einstein-Gauss-Bonnet (EGB) theory (Liko and Booth 2007; Liko 200 ) in higher dimen­

sions. There are of course many more theories of gravity in higher dimensions. Some of

the modern approaches in five dimensions incorporating a large extra dimension include

braneworld cosmology (Brax and van de Bruck 2003; Maartens 2004) and induced-matter

theory (Overduin and Wesson 1997; Liko et al 2004).

Chapter 1. Introduction 8

The motivation for extending the IH framework specifically to the two theories presented

in this thesis came from their relevance within the context of ST. EM-CS theory is imp or-

tant in ST because, in five dimensions, the corresponding action with negative cosmological

constant is the bosonic action of N = 1 gauged supergravity (Cremmer 1980); black holes in

particular are described by solutions to the bosonic equations of motion with all fermionic

fields and their variations vanishing in the vacuum (Gibbons et al 1994; Gauntlett et al

1999; Gutowski and Reali 2004). In addition, the action in four dimensions reduces to the

Einstein-Maxwell (EM) action, which is the bosonic action of N = 2 gauged supergravity

(Gibbons et al 1994). EGB theory is important in ST because the corresponding action

contains the only possible combination of curvature-squared interactions for which the lin-

earized equations of motion do not contain any ghosts (Zwiebach 1985; Zumino 1986; Myers

1987). This is particularly important in ST because of the no-ghost theorem (Polchinski

1998), which states that the BRST inner product is positive.

1. 3 Overview and main results

In Chapter 2 we consider the phase space of solutions to the equations of motion for the

EM-CS action

(1.5)

Here, R is the scalar curvature, g is the determinant of the spacetime metric tensor 9ab

(a, b, ... E {0, .. . , D - 1} ), Aa is the vector potential and Fab = 8aAb - 8bAa (with F 2 =

Fabpab) is the field strength. The constants appearing in the action are the gravitational

Chapter 1. Introduction 9

coupling constant "'D = 81rG D and the cosmological constant A. The cosmological constant

is given by

£ A =

2£2 (D - 1)(D - 2), (1.6)

where£ E { - 1, 1} and Lis the (anti-)de Sitter radius. Also, we set c = li = 1 from here on

unless otherwise stated. The last term is a Chern-Simons (CS) term for the electromagnetic

field; here A = 0 if D is even and A = 1 if D is odd. The field equations that are derived

from the action (1.5) when the metric is varied are the Einstein equations

Gab = 2Tab - A gab (1.7)

with the Einstein tensor Gab and stress-energy tensor Tab given by

(1.8)

The field equations that are derived from the action (1.10) when the vector potential is

varied are the Maxwell-Chern-Simons equations

\1 pab = 4(D + 1)A iCJ ···CD- ! F ... F a 3J3A CJC2 CD - 2CD - 1 •

(1.9)

There are several solutions to these equations that describe black holes. In four dimensions

with A = 0 the equations are solved by a family of topological Kerr- ewman-ADS (K -

ADS) spacetimes (Kostelecky and Perry 1996; Caldarelli and Klemm 1999). The solutions

that are supersymmetric describe: (a) rotating and extremal black holes with horizon cross

sections of spherical, cylindrical or toroidal topologies and having non-trivial electromag-

netic fields; and (b) non-rotating and extremal black holes with constant curvature horizon

cross sections of genus g > 1 and with magnetic (but not electric) charge. In five dimensions,

Chapter 1. Introduction 10

with A = 0, the simplest solution is the five-dimensional Reissner-Nordstri:im (RN) space­

time (Tangherlini 1963; Myers and Perry 19 6). The equations admit two asymptotically

flat solutions that describe supersymmetric black holes. These are the Br ckenridge-Myer -

Peet-Vafa (BMPV) black hole (Breckenridge et al 1997), and the Elvang-Emparan-Mateos­

Reall (EEMR) black ring (Elvang et al 2004). The Gutowski-Reali (GR) black hole is a

generalization to ADS spacetime of the BMPV black hole (Gutowski and Reali 2004). The

main purpose of the work in (Booth and Liko 2008; Liko and Booth 2008) was to develop

a quasilocal framework for these black holes.

First, we examine the boundary conditions and their con equences. To this end, we

consider the action (1.5) in the first-order connection formulation of general relativity, after

which we specify the boundary conditions that are imposed onto the inner boundary of

M . These boundary conditions capture the notion of a weakly isolat d horizon (WIH) that

physically corre ponds to an isolated black hole in a surrounding spacetime with (possibly

dynamical) fields and leads to the zeroth law of black-hole mechanics.

Next we investigate the mechanics of the WIHs. We show that the action principle

with boundaries is well defined by explicitly showing that the first variation of the surface

term vanishes on the horizon. We then find an expression for the symplectic structure by

integrating over a spacelike (D - I)-surface the antisymmetrized second variation of the

surface term and adding to this the pullback of the resulting two-form to the WIH. This

allows us to find an expression for the local version of the (equilibrium) first law of black-hole

mechanics in dimensions D ~ 5.

Summarizing thus far, we have the following:

R esult 1. A charged and rotating WIH 6. C M on the phase space of solutions of EM-CS

Chapter 1. Introduction 11

theory in D dimensions satisfies the zeroth and first laws of black-hole mechanics.

After proving that the first law holds, we restrict our study to the stronger notion of

(fully) IHs. These are WIHs for which the extrinsic as well as intrinsic geometries are

invariant under time translations . . For these horizons, the sign of the surface gravity "-(t)

is well defined. The requirement that "-(t) 2: 0 therefore allows us to define a parameter

that provides a constraint on the topology of the IHs. We find that the integral of the

scalar curvature of the cross sections of the IH (in a spacetime with nonnegative cosmo­

logical constant) have to be strictly positive if the dominant energy condition is satified.

Furthermore, this integral will be zero if the horizon is extremal and non-rotating, and the

stress-energy tensor Tab is of the form such that Tab!!anb = 0 for any two null vectors !! and

n with normalization !!ana = - 1 at the horizon. For negative cosmological constant there

is no restriction on the scalar curvature of the cross sections of the IH.

Summarizing now, we have the following:

Result 2. The IH cross sections in a higher-dimensional spacetime with nonnegative cos­

mological constant are of positive Yamabe type; if A < 0 then there is no restriction on the

sign of the scalar curvature.

This result is in agreement with recent work on the topological constraints of higher­

dimensional black holes in globally stationary spacetimes (Helfgott et al 2006; Galloway

2006; Galloway and Schoen 2006). We note that the physical content of the stress-energy

tensor at this point is completely arbitrary. Therefore Result 2 implies that the topology

considerations are valid for any matter (nonminimally) coupled to Einstein gravity. In the

case of electromagnetic fields with or without the CS term, the scalar Tab!!anb is the square

of the electric flux crossing the horizon.

Chapter 1. Introduction 12

In Chapter 3 we examine the restrictions that are imposed on IHs if one assumes that

they are supersymmetric. To do this we specialize to IHs in four dimensions with negative

cosmological constant and in five dimensions with vanishing cosmological constant. The

former theory is the bosonic part of four-dimensional N = 2 gauged supergravity, and the

latter theory is the bosonic part of five-dimensional N = 1 supergravity. We show that

the existence of a Killing spinor in four dimensions requires that the induced (normal)

connection w on the horizon has to be non-zero unless the electric charge (but not magnetic

charge) vanishes, and that the surface gravity"' has to be zero. The former condition means

that the gravitational component of the horizon angular momentum is non-zero provided

that w is not a closed one-form. The latter condition means that the IH is extremal.

Likewise, we show that the existence of a Killing spinor in five dimensions requires that w

vanishes and this immediately also gives "'= 0.

Summarizing now, we have the following:

Result 3. A SIH of four-dimensional N = 2 gauged supergmvity is extremal, and is

either: (a) rotating with non-trivial electromagnetic field; or {b) non-rotating with constant

curvature horizon cross sections and magnetic (but not electric) charge. A SIH of four­

dimensional N = 2 supergmvity and of five-dimensional N = 1 supergmvity with zero

cosmological constant is non-rotating and extremal.

The topological KN-ADS family of solutions (Caldarelli and Klemm 1999) describe rotat­

ing and extremal supersymmetric black holes in four-dimensional ADS spacetime. Among

the special cases is a solution describing a non-rotating and extremal black hole with con­

stant curvature horizon cross sections and magnetk charge. The BMPV solution describes

an extremal black hole with nonvanishing angular momentum and non-rotating Killing

Chapter 1. Introduction 13

horizon; this black hole solution is an example of a distorted IH with arbitrary rotations

in the bulk fields (Ashtekar et al 2004). When the angular momentum vanishes this solu-

tion reduces to the extremal RN solution in isotropic coordinates. The conclusions drawn

from our Result 2 together with Result 3 are that the only possible horizon topologies for

SIRs are 82 in four dimensions (when A = 0) or 83 and 8 1 x 8 2 in five dimensions. Both

these topologies have been realized and the corresponding solutions, for example the BMPV

black hole (Breckenridge et al 1997) and the EEMR black ring (Elvang et al 2004), are well

known. The torus topology is a special case that can occur only if the stress-energy tensor

is of the form such that Tab.eanb = 0 for any two null vectors .e and n with normalization

fana = - 1. A solution describing such a black hole has yet to be discovered.

In Chapter 4 we consider the phase space of solutions to the equations of motion for the

EGB action

(1.10)

In addition to the quantities that also appear in the EM-CS action (1.5), the action (1.10)

also contains explicit dependence on the Riemann tensor Rabcd and Ricci tensor Rab = R c acb·

The constant a is the GB parameter. The field equations that are derived from the action

(1.10) when the metric is varied are the EGB equations

Go> ~ - Ago> + a{ ~ ( R2- 4RaJR'' + R,.,JR"''!) 9o>

2RRo> + 4R~R,' + 4Jl.MR"' - 2R.,R,""'} . (1.11)

When a = 0 these equations reduce to the vacuum Einstein equations. There are several

solutions to these equations that describe black holes. The simplest were found indepen-

dently by Boulware and Deser (1985) and by Wheeler (1986a; 1986b), and describe a static

Chapter 1. Introduction 14

spherically symmetric black hole. The causal structure and thermodynamics of this solution

were later studied by Myers and Simon (1988). The solution was subsequently extended to

ADS spacetime by Cai (2002) and by Cho and Neupane (2002). The purpose of the work

in (Liko and Booth 2007; Liko 2008) was to develop a quasilocal framework for these black

holes.

Just as for IHs in EM-CS theory, we begin by examining the boundary conditions and

their consequences. It turns out that for the zeroth law to be satisfied, the boundary

conditions need to be slightly modified. Specifically, an analogue of the dominant energy

condition has to be imposed onto the Ricci tensor instead of the matter stress-energy tensor.

The zeroth law then follows naturally from the modified boundary conditions.

Next we investigate the mechanics of the WIHs. In particular, we show that the ac­

tion principle for WIHs in EGB theory is well defined by explicitly showing that the first

variation of the surface term vanishes on the horizon. This turns out to be quite compli­

cated due to the presence of the GB term. Nevertheless, we verify the differentiability of

the action for EGB theory by brute force at the expense of restricting the phase space to

non-rotating WIHs. We then find an expression for the symplectic structure by integrating

over a spacelike ( D - 1 )-surface the antisymmetrized second variation of the surface term

and adding to this the pullback of the resulting two-form to the WIH. This allows us to

find an expression for the local version of the (equilibrium) first law of black-hole mechan­

ics in dimensions D ~ 5, with an entropy expression that contains a correction term that

is proportional to the surface integral of the scalar curvature of the cross sections of the

horizon. We demom;trate the validity of our expression for the quasilocal entropy of WIHs

by directly comparing it to those expressions that are obtained by the Euclidean (Cai 2002;

Chapter 1. Introduction 15

Cho and Neupane 2002) and Noether charge (Clunan et al 2004) methods.

Summarizing thus far, we have the following:

Result 4. A non-rotating WIH .6. C M on the phase space of solutions of EGB theory in

D dimensions satisfies the zeroth and first laws of black-hole mechanics.

We conclude our investigation of IHs in EGB theory by looking at physical consequences

of the correction term in the entropy can have on the area-increase law. In order to make

the analysis concrete, the calculation is done for black holes in four dimensions, specifically

for the merging of two Schwarzschild black holes in fiat spacetime. It turns out that for

this very special case the second law of black-hole mechanics will be violated if a is greater

than the product of the masses of the black holes before merging minus a small correction

due to radiation that may be lost by gravitational waves during the merging process.

Summarizing now, we have the following:

Result 5. There is a lower bound on a for which the area-increase law will be violated

when two black holes merge.

The calculation of the bound on a is done in four dimensions. However, a similar bound

may presumably be derived for specific solutions in higher dimensions as well [although in

this case the topologies are not as severely restricted as they are in four dimensions, even

for Einstein gravity with A= 0 (Helfgott et al 2006; Galloway 2006; Galloway and Schoen

2006)]. Result 2 also corrects a long-held misconception about the GB term, namely that

its presence in four dimensions does not lead to any physical effects because the term is a

topological invariant and does not show up in the equations of motion.

In Chapter 5 we conclude the thesis with a brief summary of the work that has been

done here, and discuss some classical applications of IHs in EM-CS theory and EGB theory.

Isolated Horizons 1n EM -CS Theory

"The beginner . . . should not be discoumged if . . . he finds that he does not have the prereq-

uisite for reading the prerequisites. " '"" P Halmos

2.1 First-order action for EM-CS theory

For application to IHs, we work with the "connection-dynamics" formulation of general

relativity. For details we refer the reader to the review (Ashtekar and Lewandowski 2004)

and references therein. In this formulation, the configuration space consists of the triple

(ei, AI J> A) ; the coframe ei = e/ dxa (I , J, . .. E {0, .. . , D - 1}) determines the spacetime

metric

(2.1)

the gravitational (SO(D - 1, 1)) connection AI J = A/ Jdxa determines the curvature two-

form

(2.2)

16

Chapter 2. Isolated Horizons in EM-CS Theory 17

and the electromagnetic (U(1)) connection A determines the curvature

F = dA . (2.3)

In this thesis, spacetime indices a, b, . .. are raised and lowered using the metric 9ab, while

internal Lorentz indices I , J, ... are raised and lowered using the Minkowski metric TJIJ =

diag( -1, 1, ... , 1). The curvature n defines the Riemann tensor R1 J K L [with the convention

of Wald (1984)] via

(2.4)

The Ricci tensor is then R1 J = R~ K J , and the Ricci scalar is R = ry1 J R1 J. The gauge

covariant derivative ~ acts on generic fields WI J such that

(2.5)

The coframe defines the (D - m)-form

1 E I - € [ I I I e1

m+l 1\ .. . 1\ e10 [! ··· m - ( D - m)! I ··· m m + l ·· · D ,

(2.6)

where the totally antisymmetric Levi-Civita tensor €[1 ... ! 0 is related to the spacetime volume

element by

€ _ € e h . . . e I o aJ ... ao - l J ... Io a 1 ao · (2.7)

In this configuration space, the action (1.5) for EM-CS theory on the manifold (M, 9ab)

(assumed for the moment to have no boundaries) is given by

S =- Eu 1\ n - 2A€ - - F 1\ *F - - A 1\ F - . 1 1 I J 1 2.A (D 1)/2

21'i.D M 4 3\1'3 (2.8)

Here € = e0 1\ . . . 1\ eD - 1 is the spacetime volume element and "*" denotes the Hodge dual .

Chapter 2. Isolated Horizons in EM-CS Theory 18

The equations of motion are given by 68 = 0, where 6 is the first variation; i.e. the

stationary points of the action. For this configuration space the equations of motion are

derived from independently varying the action with respect to the fields (e, A, A). To get

the equation of motion for the coframe we note the identity

(2.9)

This leads to

(2.10)

where ~ denotes the electromagnetic stress-energy (D - 1)-form. T he equation of motion

for the connection A is

(2.11)

this equation says that the torsion T 1 Pe1 is zero. The equation of motion for the

connection A is

(2.12)

The second term in this equation is the contribution due to the CS term in the action. In

even dimensions the equation reduces to the standard Maxwell equation d * F = 0. The

equations (2.10) and (2.11) are equivalent to the field equations (1.7) and (1.9) in the metric

formulation, with the components of ~ identified with the electromagnetic stress-energy

tensor.

Chapter 2. Isolated Horizons in EM-CS Theory 19

§

M

Figure 2.1: The spacetime manifold M and its boundaries. The region of t he D-dimensional space­

time M being considered has an internal boundary 6. representing the event horizon, and is bounded

by two (D- I)-dimensional spacelike hypersurfaces M± which extend from the inner boundary 6.

to the boundary at infinity 88. M is a partial Cauchy surface that intersects 6. in a compact

(D - 2)-space §.

2.2 Boundary conditions

Let us from here on consider the manifold (M , 9ab) to contain boundaries; the conditions

that we will impose on the inner boundary will capture the notion of an isolated black hole

that is in local equilibrium with its (possibly) dynamic surroundings. We follow the general

recipe that was developed in (Ashtekar et al 2000c).

First we give some general comments about the structure of the manifold. Specifically,

M is a D-dimensional Lorentzian manifold with topology R x M, contains a (D - I)­

dimensional null surface 6. as inner boundary (representing the horizon), and is bounded

by (D-1)-dimensional spacelike manifolds M± that extend from 6. to infinity. The topology

of 6. is R x §D- 2, with §D- 2 a compact (D - 2)-space. M is a partial Cauchy surface such

that §D- 2 ~ 6. n M. See Figure 1.

The outer boundary !!A is some arbitrary (D- 1)-dimensional surface. With the exception

Chapter 2. Isolated Horizons in EM-CS Theory 20

of §2.6, we consider the purely quasilocal case in this chapter and neglect any subleties that

are associated with the outer boundary. Including this contribution in the phase space

amounts to imposing fall-off conditions on the fields for fixed A [e.g. asymptotically flat

(Ashtekar et al 2000c) or asymptotically ADS (Ashtekar et al 2007)] as they approach ~.

In §2.6 we briefly discuss rotation in asymptotically ADS spacetimes.

b. is a WIH, which is defined in the following way:

Definition I. A WIH b. is a null surface and has a degenemte metric Qab with signature

0 + ... + (with D- 2 non-degenemte spatial directions) along with an equivalence class of

null normals [£] (defined by £ "' £' <=> £' = z£ for some constant z ) such that the following

conditions hold: (a) the expansion B(t.) of fa vanishes on b.; (b) the field equations hold on

b.; {c) the stress-energy tensor is such that the vector - Tabeb is a future-directed and causal

vector; (d) £ewa = 0 and £ed = 0 for all£ E [£] (see below).

The first three conditions determine the intrinsic geometry of b.. Since £ is normal to

b. the associated null congruence is necessarily twist-free and geodesic. By condition (a)

that congruence is non-expanding. Then the Raychaudhuri equation implies that Tabea fb =

- O'abO'ab, with O'ab the shear tensor, and applying the energy condition (c) we find that

O'ab = 0. Thus, together these conditions tell us that the intrinsic geometry of b. is "time-

independent" in the sense that all of its (two-dimensional) cross sections have identical

intrinsic geometries.

Next, the vanishing of the expansion, twist and shear imply that (Ashtekar et al 2000c)

(2.13)

with "~" denoting equality restricted to b. and the underarrow indicating pull-back to

Chapter 2. Isolated Horizons in EM-CS Theory 21

b.. Thus the one-form w is the natural connection (in the normal bundle) induced on the

horizon. These conditions also imply that (Ashtekar et al 2000c)

{_JF = 0 . (2.14)

With the field equations (2.12) and the Bianchi identity dF = 0, it then follows that

(2.15)

This implies that the electric charge is independent of the choice of cross sections §D- 2

( Ashtekar et al 2000b). Similarly (in four dimensions) the magnetic charge is also a constant.

From (2.13) we find that

(2.16)

and define the surface gravity K.(i) = f_jw as the inaffi.nity of this geodesic congruence. Note

that it is certainly dependent on specific element of [f] as under the transformation f -+ zf:

(2.17)

In addition to the surface gravity, we also define the electromagnetic scalar potential <I>(e) =

-f_JA for each f E [f] and this has a similar dependence.

Now, it turns out that if the first three conditions hold, then one can always find an

equivalence class [f] such that (d) also holds. Hence this last condition does not further

restrict the geometries under discussion, but only the scalings of the null normal. However,

making such a choice ensures that (Ashtekar et al 2000c):

(2.18)

Chapter 2. Isolated Horizons in EM-CS Theory 22

These conditions follow from the Cartan identity, (2.14) and the property that dJ.v is pro-

portional to € [defined below in (2.34)] (Ashtekar et al 2000c). This establishes the zeroth

law of WIH mechanics: the surface gravity and scalar potential are constant on 6..

2.3 Variation of the boundary term

Let us now look at the variation of the action (1.10). Denoting the triple (e, A, A) collec-

tively as a generic field variable 1}! , the first variation gives

8S = -2

1 1 E[w]81l! - -1- { J[w, 81l!] .

K.D M 2K.D laM

Here E[w] = 0 symbolically denotes the equations of motion and

J [w, aw] = '£1 J A 8A11 - ~A 8A

is the surface term with (D - 2)-form

~ = *F _ 4(D + 1)A A 1\ F(D-3)/ 2 .

3v'3

(2.19)

(2.20)

(2.21)

If the integral of J on the boundary 8M vanishes then the action principle is said to be

differentiable. We must show that this is the case. Because the fields are held fixed at

M± and at ~, J vanishes there. Therefore it suffices to show that J vanishes at the inner

boundary 6.. To show that this is true we need to find an expression for J in terms of

'£, A and A pulled back to 6.. As for the gravitational variables, this is accomplished by

fixing an internal basis consisting of the (null) pair (.e, n) and D - 2 spacelike vectors '!9(i)

(i E {2, ... ,D - 1}) such that

(2.22)

Chapter 2. Isolated Horizons in EM-CS Theory 23

together with the conditions

f·n=-1, f·f=n·n=f·"l9(i)=n·"l9(i)=O, "19 (·) ·"19 ( ·) = 8· · ' J lJ . (2.23)

This basis represents a higher-dimensional analogue of the Newman-Penrose (NP) formalism

(Pravda et al 2004) . The coframe e/ can be decomposed in terms of the vectors in the

basis (2.22) such that

(2.24)

summation is understood over repeated spacelike indices (i,j , k etc). The pullback of the

coframe to 6. is therefore

e I ~ - l n + "19 I "19(i) f!:_ ~ a (•) a ' (2.25)

whence the (D - 2)-form

" 1 nA1.a A2 .o Ao-2 ( 1\ .o(i!) 1\ 1\ .o(io-3)) t=-IJ ::::;:: - (D- 3)! fiJAJ ... Ao- 2{. v (i!) ... -v (io- 3) n -v . . . -v

+ 1 .o A1 .o Ao- 2 (.o(iJ) 1\ 1\ .o(io- 2)) (D- 2)!fiJAJ ... Ao-2"V(i!) . .. -v(io-2) "V • • • ·v .

(2.26)

To find the pull-back of A we first note that

\7 a £1 ::::;:: V !!:- ( eb If.b) +--

::::;:: (V a eb I )f.b + i 1 V a f b +-- +--

::::;:: eb Iwafb

(2.27)

where we used "Vaeb 1 = 0 in going from the second to the third line (a consequence of the

metric compatibility of the connection). Then, taking the covariant derivative of e acting

Chapter 2. Isolated Horizons in EM-CS Theory 24

on internal indices gives

(2.28)

with 8 representing a flat derivative operator that is compatible with the internal coframe

on 6.. Thus Oaf! ~ 0 and

(2.29)

Putting this together with (2.27) we have that

(2.30)

and this implies that the pull-back of A to the horizon is of the form

A IJ ~ -2o[InJlw + a(i) o[Lo J] + b(ij ).o [I.o J] /!:... ~ {. a a {. "U (i) a ·v(i) "U(j ) ' (2.31)

where the a~i) and b~ij) are one-forms in the cotangent space T*(t:l ). It follows that the

variation of (2.31) is

(2.32)

Finally, by direct calculation, it can be shown that the gravitational part Jcrav of the surface

term (2.20) reduces to

Jcrav[W, bw] ~ € 1\ bw. (2.33)

Here,

f = iJ(l) 1\ . .. 1\ i)(D-2) (2.34)

is the area element of the cross sections §D- 2 of the horizon.

-------

Chapter 2. Isolated Horizons in EM-CS Theory 25

Now we make use of the fact that, because I! is normal to the surface, its variation will

also be normal to the surface. That is, 81! ex: I! for some I! fixed in [I!]. This together with

£ew = 0 then implies that £eow = 0. However, w is held fixed on M± which means that

ow = 0 on the initial and final cross-sections of~ (i.e. on M- n ~ and on M+ n ~), and

because ow is Lie dragged on ~ it follows that Jcrav ~ 0. The same argument also holds

for the electromagnetic part JEM of the surface term (2.20). In particular, because the

electromagnetic field is in a gauge adapted to the horizon, £ee_ = 0, and with 01! ex: I! we

also have that £eoA = 0. This is sufficient to show that JEM ~ 0 as well. Therefore the f-

surface term JlaM = 0 for the Einstein-Maxwell theory with electromagnetic CS term, and

we conclude that the equations of motion E[w] = 0 follow from the action principle oS = 0.

2.4 Covariant phase space

The derivation of the first law involves two steps. First we need to find the symplectic struc-

ture on the covariant phase space r consisting of solutions (e, A, A ) to the field equations

(2.10), (2.11) and (2.12) on M. Once we have a suitable (closed and conserved) symplectic

two-form, we then need to specify an evolution vector field ~a. In this section we derive the

symplectic two-form. In the next section we will specify the evolution vector field which

will also serve to introduce an appropriate notion of horizon angular momentum.

The antisymmetrized second variation of the surface term gives the symplectic current,

and integrating over a spacelike hypersurface M gives the symplectic structure n = 0 ( o1, o2)

(with the choice of M being arbitrary). Following (Ashtekar et al 2000c), we find that the

Chapter 2. Isolated Horizons in EM-CS Theory 26

second variation of the surface term (2.20) gives

(2.35)

Whence integrating over M defines the bulk symplectic structure

(2.36)

We also need to find the pull-back of J to ~ and add the integral of this term to nB so

that the resulting symplectic structure on r is conserved. If we define potentials '1/J and x

for the surface gravity "'(£) and electric potential ~(l) such that

(2.37)

then the pullback to ~ of the symplectic structure will be a total derivative; using the

Stokes theorem this term becomes an integral over the cross sections §D- 2 of~. Hence the

full symplectic structure is given by

-1- f [<hEu 1\ 02A1

J - 02Eu 1\ 01A1J - 01 <P 1\ 02A + 02<P 1\ 01A] 21'\,D j M

+~ J [61€ 1\ 62'1/J - 62€ 1\ 61 '1/J + 01 <P 1\ 02X - 02 <P 1\ 01XJ . "'D fsv -2

2.5 Angular momentum and the first law

(2.38)

In D dimensions, there are l (D - 1)/2J rotation parameters given by the Casimir invari-

ants of the rotation group SO(D - 1). Here, "l·J" denotes the "integer value of". For a

Chapter 2. Isolated Horizons in EM-CS Theory 27

multidimensional WIH rotating with angular velocities nL (~ = 1, ... , l(D - 1)/2J), a suit-

able evolution vector field on the covariant phase space is given by (Ashtekar et al 2001;

Ashtekar et al 2007)

L(D-1)/2J

~a = zea + L Dd)~ . (2.39) L=l

Here, ¢~ are spacelike rotational vector fields that satisfy

(2.40)

The vector field ~ is similar to the linear combination ( = t + L:L DLmL (with t a timelike

Killing vector and mL spacelike Killing vectors) for the KN solution. By contrast, we note

that~ is spacelike in general and becomes null when all angular momenta are zero, while (

is null in general and becomes timelike when all angular momenta are zero.

Moving on, the first law now follows directly from evaluating the symplectic structure

at (8, 8d (Ashtekar et al 2007). This gives two surface terms: one at infinity (which is

identified with the ADM energy), and one at the horizon. We find that the surface term at

the horizon is given by

(2.41)

where we used K:(ze) = £ze'I/J = d..Jw and <l>(ze) = £zeX = zt'..JA. These potentials are

constant for any given horizon, but in general vary across the phase space from one point

to another. This implies that (2.41) is not in general a total variation. However, if K:(ze) ,

<I> (ze) and nL can be expressed as functions of the surface area A , charge Q and angular

Chapter 2. Isolated Horizons in EM-CS Theory

momenta .J, defined by

s

Q

and satisfy the integrability conditions

an a"' aA' aQ

acp an aA' aQ

then there exists a function£ such that (Ashtekar et al 2001; Ashtekar et al 2007)

In this case (2.41) becomes

28

(2.42)

(2.43)

(2.44)

(2.45)

(2.46)

(2.47)

which is the first law (for a quasi-static process). Therefore WIHs in D-dimensional EM-

CS theory satisfy the first law (and the zeroth law) of black-hole mechanics. This is in

agreement with (Gauntlett et al 1999), but with a very important difference. Here, all the

quantities appearing in the first law are defined at the horizon; no reference was made to

the boundary at infinity.

Remarks. Several remarks are in order here.

1. The expression (2.44) implies that the horizon angular momentum contains contribu-

tions from both gravitational and electromagnetic fields, here referred to as :TGrav and

:TEM. This is in contrast to the standard angular momentum expressions at infinity,

Chapter 2. Isolated Horizons in EM-CS Theory 29

such as the Komar expression. One can show (Ashtekar et al 2001; Ashtekar et al

2007) that JGrav is equivalent to the (quasilocal) Komar integral

JK = --- *d¢ 1 i 81rGD so- 2 '

(2.48)

and this matches the expression for Killing horizons at infinity.

2. It would appear that if JGrav = 0 then there is still a non-zero contribution to (2.44)

from JEM · However, it can be shown (Ashtekar et al 2001) that if¢ is the restriction

to 1::1 of a global rotational Killing field r.p contained in M, then JEM is actually the

angular momentum of the electromagnetic radiation in the bulk. What happens is

that the bulk integral f M Tabr.padsb can be written as the sum of a surface term at 1::1

and a surface term at@. Therefore we say that a non-rotating WIH is one for which

JGrav = 0.

3. The charges at@ are the charges of the spacetime and are independent of the charges

at /::1.

2.6 Rotation In ADS spacetime

Currently there is a lot of interest in the ADS/CFT correspondence (Maldacena 1998;

Witten 1998a; Witten 1998b; Aharony et al 2000). A significant amount of effort on the

gravity side has been focused on finding charged and rotating black hole solutions in five-

dimensional ADS spacetime, both non-extremal in general (Hawking et al 1999; Chamblin

et al 1999; Hawking and Realll999; Klemm and Sabra 2001 ; Cvetic et al 2004; Chong et al

2005) and supersymmetric in particular (Gutowski et al 2004; Kunduri et al 2006; Kunduri

et al 2007a; Kunduri and Lucietti 2007).

Chapter 2. Isolated Horizons in EM-CS Theory 30

For these black holes, however, there is an ambiguity in how the conserved charges are

defined. This was first pointed out by Caldarelli et al (2000). The ambiguity arises because

for rotating black holes in ADS spacetime there are two distinct natural choices for the

timelike Killing field. To see this, let's compare the KN solution (with A = 0) and the KN­

ADS solution. The KN solution contains the vector K = 8/8t with which one can define

the charges. When A < 0, however, there is another timelike Killing vector in addition to

K that appears and is given by K' = 8/0t + (a/L2 )8/8¢. For the KN-ADS solution, K

remains timelike everywhere outside the event horizon which implies that if K is chosen

as the generator of time translations then there is no ergoregion present. By contrast, K'

becomes spacelike near the event horizon which implies that if K' is chosen as the generator

of time translations then there is an ergoregion in the neighbourhood of the event horizon.

Physically, this means that defining the conserved charges with respect to K corresponds to

a frame at infinity that is co-rotating, whereas defining the conserved charges with respect

to K' corresponds to a frame at infinity that is non-rotating.

The original motivation for defining the conserved charges with respect to K was that

the corresponding boundary CFT conserved charges satisfy the first law of thermodynamics

(Hawking et al 1999); but this comes at the cost that the bulk conserved charges do not

(Caldarelli et al 2000; Gibbons et al 2005). This claim has by now been corrected. As was

shown in (Gibbons et al 2006) , one can always pass from the bulk conserved charges to the

boundary conserved charges in such a way that both sets separately satisfy the first law.

The key to this resolution is that the conserved charges of a rotating black hole in ADS

spacetime have to be measured with respect to the timelike vector which corresponds to a

frame that is non-rotating at infinity.

Chapter 2. Isolated Horizons in EM-CS Theory 31

From the above considerations, it is clear that rotation in ADS spacetime should be

independent of the coordinates that are used. This is especially crucial when considering

supersymmetric black holes in ADS spacetime (the extremal limit of a non-rotating ADS

black hole results in a naked singularity). In this section we will briefly discuss how the IH

framework provides a resolution to the above pathology.

To begin, we define an asymptotically ADS spacetime. Following (Ashtekar and Das

2000; Ashtekar et al 2007), we have the following:

Definition II. A spacetime (M, gab) is said to be asymptotically ADS if there exists a

spacetime (M, 9ab) with outer boundary f such that M- f is diffeomorphic toM and the

following conditions hold: (a) there exists a function n on M for which 9ab = 0 2gab on M;

{b) n vanishes on f but the gradient '\1 an is nowhere vanishing on f; (c) the stress-energy

tensor Tab on M is such that n - (D- 2)Tab has a smooth limit to f; and {d) the Weyl tensor

Cabcd of 9ab is such that n-(D- 4 )Cabcd is smooth on M and vanishes on f.

These are the standard boundary conditions which have been tailored to ensure that a

spacetime will be asymptotically ADS. Their meaning is discussed in detail in (Ashtekar

and Das 2000).

In the presence of a negative cosmological constant and with no matter fields, the

covariant phase space of WIHs is modified to include a set of conserved charges at f

(Ashtekar et al 2007). These are the Ashtekar-Magnon-Das (AMD) charges (Ashtekar and

Magnon 1984; Ashtekar and Das 2000)

f"il(J) L i E- ka-b-.;z = -- ab"UE: ~ 81rGD cD- 2 '

(2.49)

with ka a Killing vector field that generates a symmetry (i.e. time translation etc), u.a the

-------------------------------------

Chapter 2. Isolated Horizons in EM-CS Theory 32

unit timelike normal to c0 - 2 , e the area form on c0 - 2 and Eab the leading-order electric

part of the Weyl tensor. Explicitly we have that

E- 1 n3-DC- -c-d ab = D _ 3 ~' abcdn n , (2.50)

where iia = '\lan. As was shown in Appendix B of (Hollands et al 2005) , inclusion of

antisymmetric tensor fields in the action does not contribute anything to the charges at .Y

because the fields fall off too quickly. Therefore the charges at infinity for EM-CS theory

are precisely the AMD charges (2.49).

Gibbons et al (2005) showed that the asymptotic time translation Killing field for an

exact solution has to be chosen in such a way that the frame at infinity is non-rotating. If

this is done then the AMD charge evaluated for the solution will result in an expression

for mass that satisfies the first law. Moreover, Gibbons et al (2006) showed that using this

definition for the asymptotic time translation has to be used for a consistent transition to

the conserved charges of the boundary CFT.

Let us summarize. The IH framework provides a coherent physical picture whereby

two sets of conserved charges arise in ADS spacetime: the charges measured at infinity

and the local charges measured at the horizon. The local conserved charges at the horizon

then satisfy the first law. When evaluated on exact solutions to the field equations, the

charges at infinity correspond to asymptotic symmetries that are measured with respect to

a non-rotating frame at infinity.

The description of ADS black holes presented here is somewhat different from the de-

scription of black holes in globally stationary spacetimes where an ambiguity appears that

manifests itself as a choice of whether the conserved charges are measured with respect to

Chapter 2. Isolated Horizons in EM-CS Theory 33

a frame at infinity that is rotating or non-rotating. This ambiguity does not appear in the

IH framework essentially because the conserved charges of the black hole are measured at

the horizon, and the corresponding first law is intrinsic to the horizon with no mixture of

quantities there and at infinity!

2. 7 A topological constraint from extremality

One of the properties of an extremal black hole is that its surface gravity is zero. Another

property is that its horizons are degenerate: the inner and outer horizons coincide. As a

result, an extremal black hole is one for which there are no trapped surfaces "just inside"

the horizon. This property was recently used (Booth and Fairhurst 2008) to define an

extremality condition for quasilocal horizons. We note here the evolution equation for the

expansion of the null normal na (Booth and Fairhurst 2008):

(2.51)

Here, R is the scalar curvature of § 0 - 2, da is the covariant derivative operator that is

compatible with the metric

(2.52)

(2.53)

is the projection of w onto §D-2 . w is referred to as the rotation one-form.

Our desire is to apply the expression (2.51) to black holes, and in order to do this we need

to impose some restrictions on the WIHs. In order to proceed we now restrict our attention

Chapter 2. Isolated Horizons in EM-CS Theory 34

to fully isolated horizons (IHs). These are WIHs for which there is a scaling of the null

normals for which the commutator [.Ce, V] = 0, where'D is the intrinsic covariant derivative

on the horizon. This means that not only is condition (d) of Definition I satisfied, but also

it implies that [.Ce, 'D]na = 0 (Ashtekar et al 2002). In contrast to the condition (d) for

WIHs, then, this stronger condition cannot always be met and geometrically such horizons

not only have time-invariant intrinsic geometry, they also have time-invariant extrinsic

geometry. That said it is clear that this condition similarly fixes .e only up to a constant

scaling. As such it does not uniquely determine the value of the surface gravity "'(e) but does

fix its sign. In particular this allows us to invariantly say whether or not "'(e) vanishes. This

then gives rise to an invariant characterization of extremality that is intrinsic to the horizon:

a horizon is sub-extremal if "' > 0 (B(n) < 0) and extremal (with degenerate horizons) if

K. = 0. Further, .CeB(n) = 0 and combining this with the fact that the inward expansion B(n)

should always be less than zero, an integration of (2.51) gives

(2.54)

(Here we used - Agab.eanb = A and the fact that §50_ 2 €dawa = 0.) This inequality provides

an alternative characterization of extremal IHs: if TJ < 0 ("' > 0 and B(n) < 0) then b.

is nonextremal, and if TJ = 0 (K = 0) then b. is extremal. However, this inequality also

provides a topological constraint on the cross sections of b.. To see this, rewrite (2.54) so

that

(2.55)

Now, observe that the dominant energy condition requires that Tab fanb ~ 0. In addition,

llwll 2 is manifestly non-negative. The inequality (2.55) therefore restricts the topology of

Chapter 2. Isolated Horizons in EM-CS Theory 35

the cross sections of the horizon. The condition (2.55) is the same as the one that was found

in four dimensions for marginally trapped surfaces (Hayward 1994) , nonexpanding horizons

(Pawlowski et al 2004) and dynamical horizons (Ashtekar and Krishnan 2003; Booth and

Fairhurst 2007).

For nonextremal horizons, T/ < 0, and the constraint (2.55) splits into two possibilities,

depending on the nature of the cosmological constant:

• A ~ 0. The ·integral of the scalar curvature is strictly positive. In four dimensions

the GB theorem says that fs2 f.R = 81r(1 - g), with g the genus of the surface § 2. In

this case Tf < 0 implies that g = 0 and hence the only possibility is that the cross

sections are two-spheres S2 . In five dimensions T/ < 0 implies that the cross sections

are of positive Yamabe type; this implies that topologically § 3 can only be a finite

connected sum of the three-sphere S 3 or of the ring S 1 X S 2 (Schoen and Yau 1979;

Galloway and Schoen 2006; Galloway 2006) . Both these topologies have been realized

and the corresponding solutions, for example the Myers-Perry black hole (Myers and

Perry 1986) and the Emparan-Reall black ring (Emparan and Reall 2002), are well

known.

• A < 0. The integral of the scalar curvature can have either sign, or even vanish, and

the inequality will always be satisfied. The only restriction is that

(2.56)

There is no constraint on the topology of §D- 2 except that the space has to be

compact. Owing to this special property, many such black holes have been found with

0

exotic topologies in D ~ 3 dimensions. See e.g. (Baiiados et al 1993; Aminneborg

Chapter 2. Isolated Horizons in EM-CS Theory 36

0

et al 1996; Vanzo 1997; Aminneborg et al 1998; Baiiados 1998; Baiiados et al 1998;

Klemm et al 1998).

For extremal horizons, 7J = 0, and the constraint (2.55) becomes an equality. In this case

the same restrictions apply to fsD-2 f.R as for nonextremal horizons. However, there is also

a special case that occurs:

j f.R = O hD-2 (2.57)

for an extremal and non-rotating (w = w = 0) horizon when the scalar Tab.eanb vanishes on

the horizon. This case corresponds to the torus topology yD- 2.

Remark. Although the expression (2.56) does not constrain the topology of ADS black

holes explicitly, there is an interesting area-topology relation that comes out. The cos-

mological term can be integrated out, and upon rearranging to isolate the surface area

(2.58)

In four dimensions, the GB theorem then implies that

(2.59)

This implies that the maximum allowed angular momentum is bound by the genus and

area of the horizon; .see (Booth and Fairhurst 2008; Hennig et al 2008) for discussions of

the corresponding result for asymptotically flat spacetimes and appendix B of (Booth and

Fairhurst 2008) for a particular discussion of Kerr-ADS.

Alternatively, reversing the inequality, one can view it as bounding the allowed area

of isolated horizons from below by the scale of the cosmological curvature and the genus

Chapter 2. Isolated Horizons in EM-CS Theory 37

of the horizon: higher genus horizons necessarily have larger areas. Similar bounds have

previously been discovered for stationary ADS black holes (Gibbons 1999; Woolgar 1999;

Cai and Galloway 2001).

Supersymmetric isolated horizons

"String theorists listening to talks on loop quantum gravity are often puzzled by the lack of

interest in supersymmetry and higher dimensions, which string theory has shown seem to

be required to satisfy certain criteria for a good theory." rv L Smolin

3.1 Black holes and Killing spinors

Until now we have discussed the mechanics of WIHs in arbitrary dimensions. We now

specialize to supersymmetric horizons and in particular we focus on the bosonic sector of

four-dimensional N = 2 gauged supergravity and the bosonic sector of five-dimensional

N = 1 supergravity. In both cases, black holes are solutions to the bosonic equations

of motion and so the fermion fields vanish. By definition, supersymmetric solutions are

invariant under the full supersymmetry transformations that are generated by spinor fields.

This means that for black hole s<;>lutions, these transformations should leave the fermion

fields unchanged (and vanishing) . Therefore any such black hole solutions must admit a

Killing spinor field.

38

Chapter 3. Supersymmetric isolated horizons 39

For full stationary black hole solutions such as those discussed in (Caldarelli and Klemm

1999; Gauntlett et al 1999; Gutowski and Reali 2003), the Killing spinor gives rise to

a (timelike) time-translation Killing vector field in the region outside of the black hole

horizon. However, in the quasilocal spirit of the isolated horizon programme we will only

assume the existence of a Killing spinor on the horizon itself. In this case the spinor will

generate a null geodesic vector field that has vanishing twist, shear, and expansion and this

is an allowed .e on the WIH.

As we did in §2.7, we will consider fully IHs, which allows for a clear difference between

nonextremal and extremal IHs. Finally we define a supersymmetric isolated horizon (SIH)

as an IH on which the null vector generated by the Killing spinor coincides (up to a free

constant) with the preferred null vector field arising from the IH structure. As we shall

now see these are necessarily extremal as well as having restricted geometry, rotation, and

matter fields.

3.2 Killing spinors in four dimensions

We will first consider the four-dimensional action. With D = 4 and A = -3/ L2 the action

(2.8) is the bosonic action of N = 2 gauged supergravity. The (extremal) KN-ADS black

hole, which is a solution to the N = 2 supergravity with the fermion fields set to zero. As

was shown in (Kostelecky and Perry 1996), the condition for a supersymmetric KN-ADS

black hole in four dimensions to have positive energy is that

(3.1)

Chapter 3. Supersymmetric isolated horizons 40

which is the extremality condition for the KN-ADS black hole relating the mass 9J1, total

charge .Q = Jq~ + q~ (with Qe and Qm the electric and magnetic charges) and angular

momentum J = a9J1 at infinity. This is also the saturated Bogomol'ny-Prasad-Sommerfeld

(BPS) inequality. When A= 0 the equality (3.1) reduces to (Gibbons and Hull1982)

(3.2)

which is the extremality condition for the KN black hole.

For four-dimensional N = 2 gauged supergravity, we shall employ the conventions of

(Caldarelli and Klemm 2003). The corresponding (bosonic) action is

S = -- Eu td1 + -t::- -F 1\*F. 1 1 IJ 6 1 167rG4 M £ 2 4

(3.3)

The necessary and sufficient condition for supersymmetry with vanishing fermion fields is

that there exists a Killing spinor f.a such that

[ i D be 1 ] Va + 4rbc'Y 'Ya + £'Ya f. = 0 . (3.4)

Here, 'Ya are a set of gamma matrices that satisfy the anticommutation rule

(3.5)

and the antisymmetry product

'Yabcd = f.abcd · (3.6)

'Ya1 ... ao denotes the antisymmetrized product of D gamma matrices. The spinor f. satisfies

the reality condition

(3.7)

Chapter 3. Supersymmetric isolated horizons 41

overbar denotes complex conjugation and t denotes Hermitian conjugation.

From f one can construct five bosonic bilinears /, g, va, wa and wab = wlab[ where

These are inter-related by several algebraic relations (from the Fierz identities) and differ-

ential equations (from the Killing equation (3.4)) (Caldarelli and Klemm 2003). For our

purposes the significant ones are:

VaVa - WaWa = - (!2 + l), (3.9)

vawa = 0 , (3.10)

gWa = wabvb, (3.11)

f'l!ab vewd + 1 w ed -Eabcd 29fabed , (3.12)

"ilaf FabVb, (3.13)

'\7 a9 1 1 b ed

-LWa- 2EabedV F (3.14)

"ilaVb 1 J g ped LWab- Fab + 2,fabcd , (3.15)

'\7 aWb 9 F e wde + 1 p edwef d - £9ab - (a fb)ede 49abfedeJ an (3.16)

'\7 eWab ~9e[a Yb] + 2F[adfb]dee w e+ Fe dfdabe w e+ 9e[aEbJdef Wd p ef . (3.17)

These are general relations for the existence of a Killing spinor in spacetime. Although the

Killing spinor may exist in a neighbourhood of the horizon, we only require that it exist

on the horizon itself. Henceforth we specialize by setting f = g = 0 and at the same time

require that the relations hold on 6. Thus, the differential equations (3.13)-(3.17) are only

required to hold when the derivatives are pulled-back onto the horizon.

With f = g = 0, equation (3.9) implies that va and wa are both null. On an SIH

Chapter 3. Supersymmetric isolated horizons 42

we identify .ea = va and so condition (2.13) together with the differential constraint (3.15)

implies that

(3.18)

and using the skew-symmetry of Wab we can write

(3.19)

Then by equation (3.11)

.e_jw = 0 {::} "-(£) = 0. (3.20)

Thus, an SIR is necessarily extremal.

For ease of presentation we now assume that the SIR is foliated into spacelike two-

surfaces §v. One can always construct such a foliation (and its labelling) so that the

associated null normal n = dv satisfies .e_jn = - 1 (Ashtekar et al 2002). Then the two-

metric on the §v is given by (2.52) and area form on the §v can be written as

- oa b f.cd = -{; n f.abcd . (3.21)

Now we note that with"-(£) = 0 it follows from (2.53) that Wa = Wa and hence Wa E T*(§v ) ·

Finally, with respect to this foliation, the usual restriction (2.14) and (redundantly) equation

(3.13) implies that the electromagnetic field takes the form

(3.22)

on .6.. Here, E1_ and B1_ are the electric and magnetic fluxes through the surface and

j(a E T(§v) describes flows of electromagnetic radiation along (but not through) the horizon.

Chapter 3. Supersymmetric isolated horizons 43

With these preliminaries in hand we can consider the properties of SIHs in asymptotically

ADS spacetimes. First, relations (3.10) and (3.12) tell us that

(3.23)

for some function (3 (the factor of L has been included for later convenience). Then the pull­

back of (3.14) trivially vanishes without giving us any new information but (3.16) provides

a differential equation for (3 on each §v

(3.24)

where da is the intrinsic covariant derivative on §v, along with its time-invariance: £1.(3 = 0.

Next applying the various properties of extremal IHs, one can show that the pull-back

of (3.17) is

\1 ::_ Wab = 2£ (;2 - fJB1.) Qc[at'b] + 2Lf3EJ.Ec[at'b],

and combining this with (3.19) we find that

dawb + wawb = ( ; 2 - fJB1.) Qab + fJE1.f.ab .

(3.25)

(3.26)

Now as was seen in (2 .44), the gravitational angular momentum associated with a rota­

tional Killing field ¢a is

(3.27)

and so a necessary condition for non-zero angular momentum is a non-vanishing rotation

one-form Wa. That said, this is not quite sufficient as it is possible for a non-vanishing </J.JW

to integrate to zero. For example, consider the case where §v has topology S2 and ¢a is a

Chapter 3. Supersymmetric isolated horizons 44

Killing field (and so divergence-free). Then for some function ( we can write ¢a = Eabdb(

and

(3.28)

Thus, for all closed rotational one-forms (dW = 0) the associated gravitational angular

momentum will vanish. As such, it is standard in the isolated horizon literature [see e.g.

( Ashtekar et al 2001)] to take dW -=f. 0 as the defining characteristic of a rotating isolated

horizon. In our case

(3.29)

and so an SIH is rotating if and only if (JE1_ -=f. 0. Thus, a rotating horizon must have a

non-trivial electromagnetic field . This is in agreement with known exact solutions: rotating

supersymmetric Kerr-Newmann-AdS black holes as well as those with cylindrical or higher

genus horizons all have non-trivial EM fields (Caldarelli and Klemm 1999).

3.3 Killing spinors in five dimensions

We will now consider the five-dimensional action. With D = 5 and A = 0 the action

(2.8) is the bosonic sector of N = 1 gauged supergravity. As in the four-dimensional EM

theory, solutions to the bosonic equations of motion require the existence of a Killing spinor

to ensure that supersymmetry is preserved. For black holes, the positive energy theorem

together with this requirement imply that the mass and charge at infinity are constrained

such that (Gibbons el al 1994)

(3.30)

Chapter 3. Supersymmetric isolated horizons 45

As can be verified, the equality (3.30) is satisfied by the (5D) extremal RN black hole (Myers

and Perry 1986) , the BMPV black hole (Breckenridge et al1997) and the EEMR black ring

(Elvang et al 2004) .

The strategy for finding supersymmetric solutions to the bosonic equations of motion

based on Killing spinors is essentially the same in five dimensions as it is in four dimensions.

However, a problem arises specifically in five dimensions - spinors satisfying certain reality

conditions cannot be consistently defined unless they come in pairs and are equipped with a

symplectic structure. For details we refer the interested reader to the excellent Les Houches

lectures by van Nieuwenhuizen (1984) .

For five-dimensional N = 1 supergravity, we shall employ the conventions of (Gauntlett

et al 2003) . The corresponding (bosonic) action is

1 1 JJ 1 2 S =--c- Eui\D. --F I\*F - MAI\FI\F . 1671' s M 4 3v3

(3.31)

The necessary and sufficient condition for supersymmetry with vanishing fermion fields is

that there exists a Killing spin or ta (a, f3, . .. E { 1, 2}) such that

(3.32)

Here, r 1 are a set of gamma matrices that satisfy the anticommutation rule

(3.33)

and the antisymmetry product

riJKLM = €JJKLM · (3.34)

r[j ... Iv denotes the antisymmetrized product of D gamma matrices. The spinors ta satisfy

Chapter 3. Supersymmetric isolated horizons 46

the reality condition

(3.35)

the second equality is the symplectic Majorana condition, where T denotes matrix transpose

and C the charge conjugation operator satisfying

(3.36)

Spinor indices are raised and lowered u ing the symplectic structure f.Q[J which is defined

such that f.I2 = f.12 = + 1.

Using f.Q we can construct bosonic bilinears F , VI and ifJCif3 = if! (Ci{J) such that

(3.37)

As in §3.2, these bilinears are inter-related by algebraic relations and differential equations.

For our present purposes we only need the following (Gauntlett et al 2003):

(3.3 )

2 1 KL M r;;FIJF + r;;f.IJKLMF V .

v3 2v3 (3.39)

For IHs F = 0 and vi is null. Then, using (2.13) together with (3.39) , and again making

the identification va = ea, we find that an IH of five-dimensional N = 1 supergravity

equipped with a null normal f will be supersymmetric if

"" n 1 I J pKL nM Va{.b~ r;;eaeb f.IJKLM {. · .._ 2v3 .._

(3.40)

It follows that the RHS in (3.40) vanishes because of the IH condition (2.14) on F and the

pullback expression (2.25) for e, and therefore that w = 0.

Chapter 3. Supersymmetric isolated horizons 47

3.4 Interpretation

In this chapter we examined the restrictions that are imposed on IHs when they are assumed

to be supersymmetric. The necessary and sufficient condition for supersymmetry in four­

dimensional ADS spacetime is that there exists a Killing spinor E that satisfies the conditions

(3.4) . For four-dimensional SIHs in asymptotically ADS spacetimes we found that the

surface gravity vanishes identically from the algebraic conditions that are implied by the

Killing spinor equation. This means that the SIHs are necessarily extremal. A further

constraint that we found for these SIHs is that the corresponding connection one-form is

generically non-zero and is not closed. Then it follows from (3.29) that these SIHs are

rotating for non-trivial electromagnetic fields . As we will see below, such a SIH can be non­

rotating if the horizon cross sections are constant curvature surfaces and there is magnetic

(but not electric) charge.

The necessary and sufficient condition for supersymmetry in five dimensions is that there

exists a Killing spinor Ea that satisfies (3.32). For five-dimensional SIHs in asymptotically

flat spacetimes we found that w vanishes identically. This implies that the corresponding

SIHs are non-rotating. The condition also implies that "' is zero. The corresponding SIHs

are therefore non-rotating and extremal.

These properties impose additional constraints on the topology of the corresponding

IHs. For SIHs in ADS spacetime there is still no constraint on the possible topologies. The

topologies become severely restricted, however, when A = 0. For SIHs in asymptotically

flat spacetimes the connection w vanishes both in four dimensions (see the appendix) and

Chapter 3. Supersymmetric isolated horizons 48

in five dimensions. In this case the topology constraint (2.55) gives

(3.41)

In four dimensions we find that there are two possibilities for the topology of a SIH:

In five dimensions we find that there are three possibilities:

Exact solutions to the field equations for the cases where § 2 ~ S2 , § 3 ~ S3 and § 3 ~ S1 x S2

are known. The torus topologies, which are classically allowed topologies, have not been

found as of yet.

As is the case for spacetimes with no cosmological constant, SIRs in ADS spacetime

have vanishing surface gravity and so are always extremal. However , in contrast to the

asymptotically fiat case, ADS SIRs in four dimensions can be either rotating or non-rotating

with strong constraints linking the rotation to the electromagnetic and Killing spinor fields.

To give a taste of their application, let us apply these constraints to the case when w = 0.

Then, the Maxwell equations along with the extremal IR conditions tell us that E1_ and

B1_ are both constant in time (.£eE1_ = .£eB1_ = 0) . In addition, the Maxwell equations

(3.42)

0 (3.43)

Chapter 3. Supersymmetric isolated horizons 49

can be projected to §v, respectively, such that

- b-cdn F. - b cMI.n F. - b cMI.n D qa q v c db- qa n c v c db+ qa n c v drbc 0 (3.44)

(3.45)

Adding these two equations together, and using the decomposition (3.22) for Fab along with

the assumption w = 0 then gives

- b codn F. - b-cdn F. 2- b codn D d B - bd E 0 qa n <. v b cd + qa q v c db+ qa n <- v drbc = a .L + Ea b .L = · (3.46)

Hence E.L and B .L are also constant on each §v . Next the supersymmetry constraint (3.26)

says that

1 f3B.L = £2 and f3E.L = 0 . (3.47)

Thus, E .L = 0 while B.L # 0 - that is, these SIRs necessarily have magnetic, but not electric,

charges. Further, applying the extremality condition (2.55):

1-n d -a - -a T. oa b 3 2 ''- = aW + WaW + ab<- n - £ 2

B1_ -~ v·

(3.48)

(3.49)

[This equation has been solved by Kunduri and Lucietti (2008) for vacuum gravity in the

context of near-horizon geometries.] It is clear that the two-dimensional Ricci curvature R

of the §v is constant in this case - unfortunately the sign of that curvature does not seem

to be determined by the equations. Consulting a listing of exact supersymmetric black hole

solutions (Caldarelli and Klemm 1999) we see that such solutions are known: specifically

there is a supersymmetric asymptotically ADS black hole in four dimensions which can be

non-rotating if the horizon cross sections have genus g > 1. As prescribed by our formalism,

these solutions have magnetic but not electric charge.

Chapter 3. Supersymmetric isolated horizons 50

The quasilocal picture that we have presented is in excellent agreement with the re­

sults that are known for stationary spacetimes (Gibbons et al 1994; Gauntlett et al 1999;

Gutowski and Reali 2004). In that context a supersymmetric black hole in a spacetime

with A= 0 (in four and in five dimensions) contains an extremal and non-rotating horizon.

Extremality is a consequence of the BPS bounds being saturated, which implies that there

exists a Killing spinor. Non-rotation is then a consequence of the fact that a vector cannot

be constructed in the neighbourhood of a supersymmetric Killing horizon that is spacelike,

as can be seen from the algebraic conditions (3.9) and (3.38). Therefore the spacetime of

such a black hole cannot contain an ergoregion, which means that the horizon must be

non-rotating. On the other hand, a supersymmetric black hole with A < 0 contains a rotat­

ing and extremal horizon (with non-trivial electromagnetic field). The rotation is required,

otherwise the extremal limit would result in a naked singularity.

In this chapter we focused on null Killing spinors that are defined at the horizon itself.

However, the results obtained here for black holes will not be affected if we assume that

the spinors are defined globally. We note that there are many other solutions with such

spinors that are defined for the entire spacetime, which do not describe black holes. These

include pp-waves (plane-fronted gravitational waves with parallel rays) - spacetimes with

vanishing expansion , shear and twist, and are a subset of a wider class of solutions that

share this property, known as Kundt spacetimes. For details, see e.g. (Stephani et al 2003).

If a spinor is defined globally then 1!., which is hypersurface orthogonal everywhere, is also

defined globally. Therefore such spacetimes can actually be foliated by SIRs (Pawlowski et

al 2004).

Chapter 3. Supersymmetric isolated horizons 51

3.5 Relation to asymptotically flat solutions

In five dimensions, there are two supersymmetric solutions with the property that the bulk

electromagnetic field contains angular momentum while the horizon is non-rotating. These

are the BMPV black hole (Breckenridge et al 1997) and the EEMR black ring (Elvang et

al 2004).

The BMPV black hole was first discovered in (Breckenridge et al 1997) as a solution to

the equations that result from the truncation of D = 6 supergravity. Later it was shown

that this spacetime is a solution to the D = 5 EM-CS theory (Gauntlett et al 1999) . As

was shown in (Reali 2003), the BMPV black hole is the unique asymptotically flat extremal

solution to EM-CS theory with horizon topology S3 . The line element of this spacetime

can be written in the following form:

(3.50)

with

~ ( dB2 + dq} + d'lj;2 + 2 cos Bd¢d'lj;) (3.51)

d'lj; + cos Bd¢ ; (3.52)

the angular coordinates have the ranges

0 ~ () < 7r ' 0 ~ ¢ < 27r ' 0 ~ 'lj; < 47r . (3.53)

The (U(l)) vector potential that solves the EM-CS equations of motion with the metric

(3.50) is

A v'3 [(1 + £..) - l (dt + JO"J) - dt] . 2 r 2 2r2

(3.54)

Chapter 3. Supersymmetric isolated horizons 52

The parameters J.L and j are related to the total mass 9J1, charge .0 and angular momentum

J at infinity via

9J1 = 37rJ.L 4G5 '

.0 = V37rJ.L 2G5 '

Here, the mass is related to the total charge via

9J1 = V3.0 2 ,

J =- 1rj . 2G5

(3.55)

(3.56)

which implies that the BPS bound is saturated. This is a typical characteristic of super-

symmetric solitons in D = 5 supergravity (Gibbons et al 1994).

The BMPV black hole has one independent rotation parameter. With Jcrav = 0 this

corresponds to a SIH with one angular momentum given by

(3.57)

The spacetime of the BMPV black hole is described by a non-rotating spherical horizon with

angular momentum stored in the electromagnetic fields. In addition, the distribution of

angular momentum of the spacetime is such that there is a non-zero fraction on the horizon

as well as a negative fraction behind the horizon (Gauntlett et al 1999). The net result is

that the horizon geometry is that of a squashed three-sphere. Within our framework, these

interesting physical properties correspond to IHs with arbitrary distortions and rotations

in their neighbourhoods. Such IHs have been studied using multipole moments (Ashtekar

et al 2004). When the angular momentum of the BMPV black hole is zero the solution

reduces to the extremal RN solution in isotropic coordinates.

The generalization of the BMPV black hole to the case where the solution has two

independent angular momentum parameters is the EEMR black ring (Elvang et al 2004).

Chapter 3. Supersymmetric isolated horizons 53

The solution describes an extremal black hole with horizon topology S1 x S2 . The solution

was discovered by Elvang et al (2004); the properties and stringy origin were studied in

detail in (Elvang et al 2005). The line element of this spacetime can be written in the

following form:

(3.58)

with

dS2 4 = R2 [~ + __!}j£_ + (y2- 1)d1/J2 + (1- x2)d¢}]

(x- y)2 1 - x2 y2 - 1 (3.59)

f -1 Q 2 2

1 + ----=-!L(x - y) - _q_(x2 - y2) 2R2 4R2

(3.60)

wt/J = - 8~2 (1- x2)[3Q- q2 (3 + x + y)] (3.61)

wq, = 32q(1+y)+8~2(1 - y2)[3Q - q2(3+x+y)]; (3.62)

the spatial coordinates have the ranges

- 1 ~ X ~ 1 , - 00 < y ~ - 1 , 0 ~ 1/J < 27T, 0 ~ </> < 27T . (3.63)

The parameters q and Q are positive constants that are proportional to the magnetic dipole

moment and total charge, and R > 0 is the radius of a circle that is parametrized by 1/J at

y = -oo. Also note that the set of coordinates (x, </>) parametrize a two-sphere. Therefore

the horizon is "blown up" to a ring (topologically S 1 x S2) in the 5D geometry. The (U(1))

vector potential that solves the EM-CS equations of motion with the metric (3.58) is

V3 [ q ] A = T J(dt + wt/J d'I/J + wq,d</>) - 2[(1 + x)d</> + (1 + y)d'I/J] . (3.64)

The parameters q and Q are related to the total mass 9J1 and total charge .Q at infinity via

.Q = V37TQ 2G5 '

(3.65)

Chapter 3. Supersymmetric isolated horizons

and to the angular momenta J t/1 and J<P at infinity via

?Tq 2 J<P = -(3Q-q) 0

8G5

54

(3.66)

Note that !))1 = ( .../3/2)D. as for the BMPV black hole. Also note that when R = 0 the

angular momenta J t/1 and J<P are equal. This would suggest that the BMPV black hole

with equal angular momenta is essentially the EEMR black ring in the limit when R = 0.

However, in this limit there is then an apparent singularity in the four-metric and also in

the connection one-form.

The EEMR black ring has two independent rotation parameters. This corresponds to a

SIH with two angular momenta given by

(3.67)

In addition, a black ring can also have dipole charges which are naturally defined on the

horizon (Astefanesei and Radu 2006; Copsey and Horowitz 2006; Emparan and Reali 2006).

For an IH with ring topology a definition for the dipole charge P can be realized by inte-

grating the electromagnetic field strength plus the CS contribution over S2:

(3.68)

Charges of this type appear in the first law for a black ring (Astefanesei and Radu 2006;

Copsey and Horowitz 2006). However, this is not the case with the first law (2.47) that we

derived. This is probably due to the fact that the dipole charges P are associated with the

presence of magnetic charge, which we have not incorporated into our current phase space.

Chapter 3. Supersyrnmetric isolated horizons 55

Appendix: an alternate formalism in four dimensions

In four dimensions, there is an alternative formalism that can be used to derive the su­

persymmetry conditions for IHs as was originally done for fiat spacetime (Liko and Booth

2008). This is a NP-type spinor formalism , in which the necessary and sufficient condi­

tion for supersymmetry with vanishing fermion fields is that there exists a Killing spinor

1/JAA' = (aA,J3A') (A,B, ... E {1 , 2} and A' ,B', .. . E {1,2}) such that (Tod 1983; Tod 1995)

\1 AA'aB + V'i¢ABJ3A'

\1 AA'J3B'- .,fiJ>A'B'aA

0

0.

(3.69)

(3.70)

Here, ¢AB is the Maxwell spinor and ¢>A'B' is its complex conjugate. These are related to

the field strength via

(3.71)

the spinor symplectic structure is defined such that t 12 = -t21 = 1. Using the spinors a

and ,6 we can define the following set of null vectors:

(3.72)

It can be shown that the vector

(3.73)

is a Killing vector; the norm of this vector is given by

IIKII = 2VV, (3.74)

where we defined the scalar V = aA~A. It follows that KAA' can be eith r timelike (referred

to as nondegenerate) when V "I 0 or null (referred to as degenerate) when V = 0.

Chapter 3. Supersymmetric isolated horizons 56

For IHs, the case of interest is the one for which the Killing spinor is null. This is a

particularly special case because V = aA!JA = 0 implies that

(3.75)

for some function K. Putting this into the conditions (3.69) and (3.70) allows one to find

an expression for the covariant derivative in terms of the spinors (Tod 1983; Tod 1995):

(3.76)

Here, ¢ is a function defined by </JAB = ¢aAaa. Let us use this form of the covariant

derivative to find the covariant derivative of the null normal f of an IH. We find that

(3.77)

This immediately implies that

(3. 78)

and with (2.13) it follows that w = 0.

Isolat d Horizons 1n EGB Theory

"Higher-derivative theories are frequently avoided because of undesirable properties, yet they

occur naturally as corrections to general relativity and cosmic strings. " "' J Z Simon

4.1 First-order action for EG B theory

As in Chapter 2, we work in the first-order connection-dynamics formulation. Here, the

configuration space consists of the pair ( e1 , A 1 J ) (with electromagnetic fields zero) . In this

configuration space, the action (1.10) for EGB theory on the manifold (M, gab) (assumed

for the moment to have no boundaries) is given by

s = -1- r EIJ 1\ niJ - 2At:. + aEuKL 1\ niJ 1\ nKL .

2K.D }M

The equation of motion for the connection is

( 4.1)

(4.2)

This equation says that, in general, there exists a non-vanishing torsion T 1 = 9e1. To see

what constraints are imposed on T , we can use the Bianchi identity 9 0/J = 0 together

57

Chapter 4. Isolated Horizons in EGB Theory 5

with the identity

(4.3)

Substituting these into equation (2.11) gives

(4.4)

In analogy with Einstein gravity, we assume directly that the torsion in ( 4.4) vanishes.

(The torsion in Einstein gravity is zero, but this is not an assumption. The condition

follows directly from the equation of motion for the connection.) To get the equation of

motion for the co-frame we note that the variation of E is given by

(4.5)

This leads to

(4.6)

The equations ( 4.2) and ( 4.6) for the connection and co-frame are equivalent to the field

equations (1.11) in the metric formulation.

4. 2 Modified boundary conditions and the zeroth law

Let us now turn to the case for which the manifold (M, 9ab) has boundaries; the region

of spacetime that we consider for EGB theory is essentially the same as that which we

considered in Chapter 2 for EM-CS theory. Namely, (M, 9ab) is aD-dimensional Lorentzian

manifold with topology R x M, contains a (D - I)-dimensional null surface 6. as inner

boundary (repre enting the horizon) , and is bounded by (D - 1)-dimen ional manifolds M±

Chapter 4. Isolated Horizons in EGB Theory 59

that extend from~ to infinity. As in Chapter 3, we consider the purely quasilocal case and

neglect any subleties that are associated with the outer boundary.

Let us now state the modification that is required for EGB theory. First, we note that

for IHs in general relativity the dominant energy condition can be imposed interchangably

onto the Ricci tensor or the stress-energy tensor. This is because the Einstein field equations

Gab = 2Tab imply that RabVavb = 2Tabvavb for any null vector va. For IHs in EGB theory

this is no longer true because Gab =f. K.DTab· [This is also the reason why the topology

constraint (4.4) is not applicable to IHs in EGB theory.] It turns out that condition (c)

of Definition I for IHs in EGB theory has to be imposed onto the Ricci tensor in order for

the shear tensor to vanish when the Raychaudhuri equation is employed. Thus we have the

following modified definition for a WIH in EGB theory:

Definition III. A WIH ~ in EGB theory is a null surface and has a degenerate metric

Qab with signature 0 + ... + (with D - 2 nondegenerate spatial directions} along with an

equivalence class of null normals [£] (defined by £ ,....., £' {=} £' = z£ for some constant z) such

that the following conditions hold: (a) the expansion B(t) of la vanishes on~; (b) the field

equations hold on~; (c) the Ricci tensor is such that the vector - R alb is a future-directed

and causal vector; (d) £ewa = 0 and £ed = 0 for all£ E [£] .

The condition (c) on the Ricci tensor is the only modification that needs to be made

for WIHs in EGB theory. In particular the zeroth law now follows just as it does for IHs

in EM-CS theory. Remarkably, the zeroth law is essentially independent of the functional

content of the action, and follows from the boundary conditions alone. This is the same

conclusion that was obtained for globally stationary spacetimes (Wald 1993; Iyer and Wald

1994; Jacobson et al 1994) .

Chapter 4. Isolated Horizons in EGB Theory 60

4.3 Variation of the boundary term

We have seen that the definition for a nonexpanding horizon needs to be modified for EGB

gravity by imposing the analogue of the dominant energy condition directly on the Ricci

tensor. In the action principle, the main modification to the formalism is the appearance of

an additional surface term. Let us therefore reconsider the action ( 4.1) but for a region of

the manifold M that is bounded by null surface b. and spacelike surfaces M± which extend

to the (arbitrary) boundary~.

Denoting the pair ( e, A) collectively as a generic field variable W, the first variation gives

oS = -1- { E[w]ow + (-

21)0

{ J [w,ow]. 2/'i,D }M /'i,D laM

(4.7)

Here E[w] = 0 symbolically denotes the equations of motion and

- IJ J[w, ow] = Eu A oA (4.8)

is the surface term, with (D - 2)-form

- KL Eu = Eu + 2et.EIJKL 1\ 0 . (4.9)

If the integral of J on the boundary 8M vanishes t hen the action principle is said to be

differentiable. We must show that this is the case. Because the fields are held fixed at M±

and at ~, J vanishes there. So we only need to show that J vanishes at the inner boundary

b. . To show that this is true we need to find an expression for J in terms of A and IS

pulled back to b.. As for EM-CS theory we do this by fixing an internal basis consisting

of the (null) pair (£, n) and D - 2 spacelike vectors "!?(i) given by (2.22) together with the

conditions (2.23).

Chapter 4. Isolated Horizons in EGB Theory 61

The pull-back of A is the same as for EM-CS theory, and is given by (2.31). For EGB

theory we also need the pull-back of the curvature n to~ ' which can be obtained either by

direct calculation from (2.31) or by construction from the definition of the Riemann tensor.

We will take the secpnd approach here. We will use the definition (2.52) of the metric on

§D-2 and the definition (2.53) of the connection projected onto §D-2.

In thinking about these quantities it is useful to keep in mind the case where ~ is

foliated into spacelike (D - 2)-surfaces §v which are labelled by a parameter v and n is

chosen to be -dv. Then ga evolves the foliation surfaces while ga and na together span

the normal bundle T j_ (§v) on which Wa is the connection. Furthermore, the '!9(i) span the

tangent bundle T(§v) and Qab is the metric tensor for the §v· Then, it is clear that ~ is

non-rotating when w = 0 provided that the rotational vector field ¢ is tangential to ~.

We now turn to the Riemann tensor with the first two indices pulled back to ~. By

definition,

(4.10)

and with the horizon identity \1 a R_b = waf.b along with the decomposition (2.53) , a few lines +-

of algebra gives

(4.11)

where da is the covariant derivative that is compatible with the metric Qab· For a weakly

isolated horizon the zeroth law ensures that daK-(e) = 0 and if the horizon is non-rotating

then w = 0 also, whence

(4.12)

Chapter 4. Isolated Horizons in EGB Theory 62

Finally, using this result and (2.52) it is straightforward to see that

R .oc .od - e- JR .oc .od ~cd·u (i) .u (j) = qa qb efcdV(i)v(j) · (4.13)

From here one can use the fact that

d d .o(i) -d-e- fn ( - g- hn .o(i)) a bVc = qa qb qc Vd qe qf vg·uh ' (4.14)

and the identity for the Riemann tensor Rabcd associated with ilab

(4.15)

along with (2.52) to show the Gauss relation

(4.16)

l!a and na. However, k~~ = (1/2)-a(e)ilab + aab, and on a non-expanding horizon both the

expansion and shear vanish. Thus for the cases in which we are interested

(4.17)

Then upon expanding the frame indices of fl~IJ in terms of the 1!1 , n 1 and -a(h' and

applying (4.12) and (4.17), it follows that on any non-rotating WIH the pull-back of the

associated curvature is

(4.18)

where Rkl ij is the Riemann tensor associated with the (D- 2) metric ilab = 9ab+ l!anb+ nal!b·

That is, given a foliation of b. into spacelike (D - 2)-surfaces, the spacelike -a(i) give an

Chapter 4. Isolated Horizons in EGB Theory 63

orthonormal basis on those surfaces and nkl ij is the corresponding curvature tensor; for a

non-expanding horizon, these quantities are independent of both the slice of the foliation

and the particular foliation itself.

To find the pull-back to b. off:, we use the decomposition (2.25), whence the (D- 2)-

form given by (2.26) and in D ~ 5 dimensions, the (D- 4)-form

~ ~ 1 fA1.a A2 .a Ao- 4 [ 1\ .a (i1) 1\ 1\ .a(io- s)] ~IJKL - (D _ 5)!EJJKLA1 ... A 0 _4 v(i l) ... v(io- s) n ·u · · · v

+ 1 .a A1 .a Ao- 4 [.a(il) 1\ 1\ .a(io- 4)] (D _ 4)!EJJKLA1 ... A 0 _ 4v(i !) oo. ·u(io - 4) v oo • u .

In four dimensions EIJKL = EJJKL· f--

(4.19)

These expressions are somewhat formidable but on combining them to find f:IJ 1\ 6A1J

there is significant simplification. The key is to note that each term includes a total con-

traction of EIJ ... I 0 . This contraction must include one copy of each of f 1, n1

, and the 73 (h

-else that term will be zero. Similarly the resulting (D - 1) form must be proportional to

n 1\ 7J(2) 1\ .. . 1\ 7J(D- l). Then ( 4.8) becomes

J[•T• .~: .T• ] - .~: 2a [ fl J .a K .a L.a A1 .a Ao- 4] '.l!,u'.l! ~€1\uW + (D - 4)! fiJI<LA 1 .. . A 0 _4 n ·u(k)v(l)v(i l) 00 'v i0 _4

xnmnkl7J(i l) 1\ 0 0 0 1\ 7J(io - 4) 1\ 7J(m)7J(n) 1\ 6w 0

(4.20)

The first and second terms respectively come from the E I J and E I J K L parts of f: 1 J , € is f-- f-- f--

the area element defined by (2.34), and we keep in mind that the horizon is non-rotating

so that Wa = - l'i.(e)na. The second term therefore also simplifies. Given that there are only

(D - 4) elements in the spacelike basis it is reasonably easy to see that this term sums over

Chapter 4. Isolated Horizons in EGB Theory 64

cases where ( m, n) and ( i, j) are the same set of indices. That is (up to a numerical factor)

the second term amounts to contracting m with i and n with j so that the full surface term

reduces to

J[w , ow] ~ €(1 + 2aR) "ow . (4.21)

It follows that J ~ 0 because ow = 0 on the initial and final cross sections of .6. (i.e. on

M- n .6. and on M+ n .6.), and because ow is Lie dragged on .6.. Therefore the surface term

J laM = 0 for EGB gravity, and we conclude that the equations of motion E[w] = 0 follow

from the action principle oS = 0.

4.4 Covariant phase space and the first law

In order to derive the first law we need to find the symplectic structure on the covariant

phase spacer consisting of solutions (e, A) to the EGB field equations on M. We find that

the second variation of the EGB surface term ( 4.8) gives

(4.22)

integrating over M defines the bulk symplectic structure

(4.23)

We also need to find the pull-back of J to .6. and add the integral of this term to OB to

determine the full symplectic structure. From (4.21) we have

(4.24)

Chapter 4. Isolated Horizons in EGB Theory 65

which is a total derivative. Now, with the definition (2.37) for '1/J , the surface symplectic

structure Os is a total derivative, and hence upon using the Stokes theorem, becomes an

integral over § 0 - 2. The full symplectic structure for EGB gravity is therefore

= -1- { [8d5u 1\ 82AJJ - 82f-JJ 1\ 81AJJ]

2K.o JM +-1 J [81 [€(1 + 2aR)]/\ 82'1/J - 82 [€(1 + 2aR)]/\ 81 '1/J] ,

K.D Jso-2

where we have absorbed the overall (irrelevant) factor of ( - 1)0 .

(4.25)

We can now proceed to derive the first law. As before, this follows upon evaluating the

symplectic structure at (8, 8d, giving a surface term at infinity (which is identified with

the ADM energy) and a surface term at the horizon. We find that the surface term at the

horizon is given by

S1 b(8, 8t) = "'(zl) 8 J €(1 + 2aR). K.D Jso-2 ( 4.26)

where we used "'(zl) = £ ze'I/J = zt.Jw . Finally, assuming that this is a total variation, i.e.

that there exists a function £ such that Ole.( 8, 8~) = 8£, we find that

8£ = "'(ze) 8 J €(1 + 2aR) . "'D Jso-2

(4.27)

This is the first law for a WIH in EGB theory. Comparing this to the standard first law

identifies the entropy as

S = G1 J €(1 + 2aR) .

4 D Jso-2 (4.2 )

Therefore WIHs in D-dimensional EGB theory satisfy the first law (and the zeroth law) of

black-hole mechanics.

Chapter 4. Isolated Horizons in EGB Theory 66

4.5 Comparison with Euclidean and Noether charge methods

The quasilocal expression for the entropy is in exact agreement with the Noether charge

expression that was derived by Clunan et al (2004). As in that approach, no assumptions

about the cross sections §D- 2 of the horizon need to be made. An important difference,

however, is that we did not assume the existence of a globally-defined Killing vector. Instead

we had to specify the existence of a time translation vector field which mimics the properties

of a Killing vector but is not defined for the entire spacetime.

In order to compare the entropy to the Euclidean method, we need to reference a black

hole solution. The EGB equations admit the following class of (static) black hole solutions

(Cai 2002; Cho and Neupane 2002):

h(r)

( ) 2 dr2 2 2

- h r dt + h(r) + r dn(k)D- 2

k

r2

( ~---8-a_A ______ B_K,_D_a_M __ __

+ ----=- 1 - 1 - + ---=-----2a (D - l)(D - 2) (D - 2)'t(k)D- 2rD- l

(4.29)

Here, 't(k)N- 1 = 1T'NI2 ;r(N/2 + 1) is the volume of an (N - I)-dimensional space sN- l =

3~)1 of constant curvature with metric dnzk)N- li k is the curvature index with k = 1

corresponding to positive constant curvature, k = - 1 corresponding to negative constant

curvature, and k = 0 corresponding to zero curvature. M is the mass of the black hole, and

a is related to the G B parameter via

a= (D - 3)(D - 4)a. (4.30)

The singular surfaces with radii r* are given by the roots to the equation h(r = r*) = 0.

We denote the event horizon by r +. The location of this surface depends on the sign of the

Chapter 4. Isolated Horizons in EGB Theory 67

cosmological constant: if A ~ 0 then the largest root r + is the event horizon, and if A > 0

then the largest root is the cosmological horizon and therefore the second largest root is the

event horizon.

The thermodynamics of the black hole is determined in the usual way using path integral

methods (Hawking 1979) . In particular, the average energy (<e) and entropy :7 are given

by

8 (<e) = -

813 (In Z) and :7 = /3(<e) + In Z, (4.31)

where In Z is the (zero-loop) partition function and /3 is the inverse temperature. The

partition function is determined via In Z = - l[g] by evaluating the Euclidean action l [g]

(in the stationary phase approximation where g are solutions to the equations of motion

8 J 1 = 0), and the inverse temperature is determined by requiring that the Euclidean

manifold does not contain any conical singularities at r + where the manifold closes up. For

the black hole solution ( 4.29) one finds that (Cai 2002; Cho and Neupane 2002)

(<e) = M and :7 = sdD-2r~-2

[ 1 + (D - 2) 2ak] . 4GD D - 4 r~

(4.32)

Here, sdN-l = 27rN/ 2 /f(N/2) is the surface area of a unit (N- 1)-sphere. This shows that

the entropy acquires a correction due to the presence of the GB term. For the solution

(4.29), the Ricci scalar is n = (D - 2)(D - 3)k/r~, and (4.28) reduces to (4.32). Our

entropy expression is therefore in agreement with the Euclidean expression as well. In our

derivation, however, the entropy ( 4.28) automatically satisfies the first law ( 4.27).

An interesting consequence of the GB term is that it is possible for black holes to have

negative entropies when 2o:R < 1. This was first discovered by Cvetic et al (2002) and later

confirmed by Clunan et al (2004). For non-rotating horizons, the first law ( 4.27) implies

Chapter 4. Isolated Horizons in EGB Theory 68

that the energy is also negative; this is not surprising, as negative-energy solutions are

possible when A < 0 (Horowitz and Myers 1999). We will now proceed to show that the

presence of the GB term also has consequences for the area-increase law during the merging

of two black holes.

4.6 Decrease of black-hole entropy In EGB theory

For any EGB black hole in D dimensions, the Ricci scalar R integrates to a constant over

§D- 2 . It is not too surprising then, that the area always increases in any physical process

involving just one black hole with an entropy of the form (4.28) (Jacobson et al 1995).

However, this will not be the case for a system with dynamical topologies such as black-

hole mergers (Witt 2007) . This is a form of topology change, which for a space with a

degenerate metric is unavoidable even in classical general relativity (Horowitz 1991). This

fact is relevant to the current problem because the entropy expression ( 4.28) holds for

Killing horizons and WIHs, both of which are null surfaces on which the induced metrics

are degenerate.

As an example, let us consider the merging of two black holes - one with mass m1

and entropy 5I = [AI+ 2aX(§I)]/4GD, the other with mass m2 and entropy 52 = [A2 +

2aX(§2)]/4GD. Here, we have defined the surface area A = fso -2 € and the correction

term X(§) = fso -2 €R. Before the black holes merge, the total entropy is

5 5I +52

1 - G [AI + A2 + 2a(X(§I) + X(§2))]. 4 D

(4.33)

Chapter 4. Isolated Horizons in EGB Theory

After the black holes merge, the total entropy of the resulting black hole is

S' = -1-[A' + 2aX(§')J.

4GD

The expressions ( 4.33) and ( 4.34) imply that S' > S if and only if

69

(4.34)

(4.35)

Without knowing the specific details of the black holes in question, nothing further can

be said about S and S', or about the upper bound (4.35). Let us therefore consider for

concreteness the simplest case - the merging of two Schwarzschild black holes in four-

dimensional flat spacetime. This is a particularly special case as the topologies are much

more restricted than could be hoped for. First, the GB theorem [see e.g. (Hatfield 1992)]

relates the correction term to the Euler characteristic x(§) via

X(§)= j €R = 47rx(§) . !s2 (4.36)

Then the Hawking topology theorem (Hawking 1972) restricts the horizon cross sections to

be two-spheres for which x(§) = 2. For a Schwarzschild black hole the correction term is

therefore X(§) = 81r. Furthermore, the surface area of a Schwarzschild black hole is related

to its mass via

(4.37)

whence the surface areas of the initial and final black-hole states are

(4.38)

Here, a small mass parameter "' 2: 0 for the surface area of the final black-hole state has

been included. This parameter corresponds to any mass that may be carried away by

Chapter 4. Isolated Horizons in EGB Theory 70

gravitational radiation during merging. With these expressions for the areas, the bound

(4.35) in terms of the masses becomes

(4.39)

Therefore the second law will be violated if a is greater than twice the product of the masses

of the black holes before merging minus a correction due to gravitational radiation.

To summarize, the validity of the second law of black-hole mechanics was examined for

a physical process in which the topology is not constant. As was shown, the correction term

appearing in the entropy ( 4.32) can lead to a violation of the second law for certain values

of the GB parameter during the merging of two black holes. The calculation was done

for two Schwarzschild black holes in four-dimensional flat spacetime. However, a similar

bound to ( 4.39) may presumably be derived for specific solutions in higher dimensions as

well [although in this case the topologies are not as severely restricted as they are in four

dimensions, even for Einstein gravity with A = 0 (Helfgott et al 2006; Galloway 2006;

Galloway and Schoen 2006)]. Incidently, the result obtained here shows that the presence

of the GB term in the action for gravity can have nontrivial physical effects even in four

dimensions, when the term is a topological invariant of the manifold. This is in sharp

contrast to the commonly held belief within the literature that the term is only significant

in spacetimes with dimension D 2': 5.

Prospects

"Try to see through fainted views. As reality disappears in a haze. A journey between eternal

walls. The senses unfold before my eyes." ,....., T G Fischer

5 .1 Summary

In this thesis we presented two extensions of the IH framework, first to EM-CS theory

and then to EGB theory in D dimensions. In particular, we derived the local version of

the zeroth and first laws of black-hole mechanics for general WIHs on the phase space of

the corresponding theories. In addition, for EM-CS theory in five dimensions and for EM

theory in four dimensions we derived the conditions that are required by supersymmetry.

We then turned to EGB theory, for which we showed that the quasilocal entropy is in

exact agreement with the expressions that are obtained by the Euclidean and Noether

charge methods. Finally, we showed that the GB term can have physical consequences in

four dimensions even though it is a topological invariant and does not contribute to the

equations of motion.

71

Chapter 5. Prospects 72

As was stated in §1.2, our intention was to employ the IH framework with suitable

extensions to higher dimensions in order to determine the generic properties of black holes

in string-inspired gravity models. A summary of the main five results are given in §1.3.

5.2 Classical applications to EM-CS and EGB theories

There are a number of classical applications of IRs to EM-CS and EGB black holes that

can be explored. Here we briefly discuss four problems that are worth investigating.

Let us first consider applications to EM-CS theory, and in particular to the corresponding

supersymmetric black holes.

• BPS bounds. The general method for deriving the BPS bound for stationary space­

times is to construct an expression for the energy of the spacetime using spinor iden­

tities and the Einstein field equations. This method was pioneered by Witten (1981)

and Nester (1981) to prove the positive energy theorem. The method was applied

in four dimensions (Gibbons and Hull 1982) and in five dimensions (Gibbons et al

1994) to calculate the BPS bounds for the corresponding spacetimes. How can one

derive these bounds for IRs? The bounds are saturated when the spinors are super­

covariantly constant, which is associated with extremality. This suggests that the

extremality condition (2.54) can be used for IRs. This is straight-forward to do for

undistorted horizons. Let us consider the four-dimensional EM theory for illustration.

Here the contraction Tabf.anb is the square of the electric flux Ej_ crossing the surface

(Ashtekar et al 2000c). For any IH this quantity is constant over S2 and can therefore

be moved outside the integral. The result can be used to relate the charge Q on the

Chapter 5. Prospects 73

horizon to its surface area A via Q = EJ.A/(47r). For the RN solution one finds that

ry = Q2 / R2 - 1 ~ 0 with R = J A/(47r) the areal radius (Booth and Fairhurst 2007b).

When the surface gravity vanishes ry = 0 and Q = R = M with M the mass. This is

the condition for supersymmetry in four dimensions. The situation is not as obvious

for distorted horizons in five dimensions. This is because the contraction Tab.eanb,

which for EM-CS theory is again the square of the electric flux, may not be constant

over the horizon cross sections in general. However, for the BMPV black hole in par­

ticular we know that the cross sections are S 3 which have constant curvature, and

therefore El is constant on b. . From here, a charge-areal radius relation follows along

the same lines as the derivation that was outlined above for the RN black hole in four

dimensions.

• Supersymmetry and horizon geometries. It was shown (Lewandowski and Pawlowski

2003) that the IH constraints for extremal IHs of four-dimensional EM theory are

satisfied iff the intrinsic geometry of the horizon coincides with that of the extremal

Kerr-Newman (KN) solution. An extension of that analysis to IHs of five-dimensional

EM-CS theory would be of interest, particularly because it would provide a method for

deriving the geometries of the corresponding extremal IHs. This would complement a

recently developed method (Astefanesei and Yavartanoo 2007; Kunduri et al 2007b;

Kunduri and Lucietti 2007) for deriving the near-horizon geometries of extremal black

holes. While speculating on the local uniqueness theorems in five dimensions we need

to keep in mind that black holes in five dimensions are much less constrained than in

four dimensions, mainly because in five dimension there are two possible topologies

Chapter 5. Prospects 74

(S3 and S1 x S2), and also because there are two independent rotation parameters. As

a consequence of this richer structure, it is possible that two distinct black holes in five

dimensions can have the same asymptotic charges (Emparan and Reali 2002). This

is a striking example of black-hole nonuniqueness in higher dimensions. Nevertheless,

uniqueness has been established for supersymmetric black holes in five dimensions

(Reali 2003). Therefore it is expected that the five-dimensional analogues of the local

uniqueness theorem of (Lewandowski and Pawlowski 2003) do exist , but for SIRs. We

also note that, while supersymmetry constrains the geometry (i.e. connection one­

form), the dominant energy condition and the Einstein field equations are still required

to constrain the topology. Therefore we expect that there should be a unique horizon

geometry for a given topology. For example, if the topology of a SIH is S 1 x S2 , then

the geometry should coincide uniquely with the induced metric and vector potential

of the EEMR black ring solution. We also expect that, if the topology is S3 , then

the geometry should coincide uniquely with that of the BMPV black hole in geneml,

and the extremal RN black hole as a limiting case when the angular momentum of

the Maxwell fields vanishes. It would be of considerable interest to try solving the

IH constraints for a SIH when Tabfanb = 0 (at the horizon); the resulting geometry

would provide the first explicit solution of a supersymmetric black hole with toroidal

topology.

Now let us consider applications to EGB theory.

• Rotation. One of the main assumptions that we made in our calculations was that

the horizons are non-rotating. It would be interesting to extend the IH phase space

Chapter 5. Prospects 75

to include rotation by relaxing the condition w = 0.

• Torsion. The formalism presented here can be further extended by including torsion.

Recall that in §5.2 we assumed T 1 = 0 directly, which became crucial when we derived

the pull-back to !::!. of the connection. However, as the equation of motion (4.2) for

A indicates, the torsion-free condition is not imposed in D 2: 5 dimensions. If the

torsion is non-zero then the pull-back to !::!. of A will not be given by (2.31). In order

to derive the modified pull-back of A in the presence of torsion we would need to find

\1 a eb 1 explicitly. In addition, the Raychaudhuri equation would be different as well, +-

and so the boundary conditions would require a more careful analysis. The effects of

torsion on IHs should therefore lead to some interesting consequences. This would be a

particularly interesting project to work out in five dimensions, for which a solution has

recently been found that describes a supersymmetric black hole (Canfora et al 2008) .

More interestingly, there is also a solution of the equations with non-zero torsion

that describes a constant-curvature black hole with entropy that is proportional to

the surface area of the inner horizon rather than the event horizon (Baiiados 1998).

This curious interchange of thermodynamic parameters, namely the outer and inner

horizons, may be a consequence of the torsion that is present in the equations of

motion. The IH framework could be employed in order to test this hypothesis.

There are many more avenues to explore. We hope that others will consider some of

them.

Black Hole Mechanics: An Overview

A.l Thermodynamics

The study of macroscopic properties of materials without knowing their internal structure is

the science of thermodynamics. This is the branch of science concerned with the dynamics

of materials where thermal effects are important. Because no reference is made to the

internal structure, the formalism involved is very general and therefore very powerful. Let us

quickly review the four laws of thermodynamics (Reif 1965; Poisson 2000) ; this will make the

connection between the laws of black-hole mechanics and the four laws of thermodynamics

apparent.

Let us now state the four laws of thermodynamics and discuss some of their physical

consequences.

• Zeroth law. If two systems are each in thermal equilibrium with a third system, then

they are in thermal equilibrium with each other. This implies that the temperature

remains constant throughout the systems that are in thermal equilibrium with each

76

Appendix A. Black Hole Mechanics: An Overview 77

other.

• First law. The internal energy%' of a system that interacts with its surroundings will

undergo a change given by

dOll = dl2 - d1f/' (A.l)

where dl2 is the amount of heat absorbed by the system and d1f/ is the amount of

work done by the system during its change of macrostate. This is really just the

statement of conservation of energy. It implies that a gain of heat to a system can do

physical work on its surroundings.

• Second law. Heat flows spontaneously from higher temperatures to lower tempera­

tures. This means that: (i) the spontaneous tendancy of a system to go toward ther­

mal equilibrium cannot be reversed without changing some organized energy (work)

into some disorganized energy (heat); (ii) it is not possible to convert heat from a hot

reservoir into work in a cyclic process without transferring some heat to a colder reser-

voir; (iii) the change in entropy dY = d.fl2 /T (with T the temperature) of a system

and its surroundings is positive and approaches zero for any process that approaches

reversibility.

• Third law. The difference in entropy o.Y between states connected by a reversible

process goes to zero as the temperature T goes to absolute zero. Unlike the other

laws which are based on classical considerations, the third law is a consequence of

quantum mechanics. The third law implies that a system at absolute zero will drop

to its lowest quantum state and thus become completely ordered.

Appendix A. Black Hole Mechanics: An Overview 78

The first law expresses the change in internal energy of a system in terms of inexact

differentials. This means that, unlike the difference d.f2 - d1f/, seperately d.f2 and d1f/

depend on the path that is taken from the initial state to final state. The first law can be

expressed in terms of exact differentials though. First, we note that in a quasi-static process

where the volume of a fluid changes but the pressure remains approximately unchanged,

the physical work done by the system is given by d1f/ = Pd"f/. Then, modulo the second

law we can write the first law in the more familiar form:

dOll = Td.Y - Pd1' . (A.2)

This is the form of the first law that appears in most references on gravitational physics

that discuss the laws of black-hole mechanics.

A.2 Black-hole mechanics: global equilibrium

Now we will review the corresponding laws of black hole mechanics. A beautiful introduction

is given in the book by Poisson (2004). The laws of black-hole mechanics were first formu­

lated for globally stationary spacetimes in four-dimensional Einstein gravity (Bardeen et al

1973), and later extended using covariant phase space methods to D-dimensional spacetimes

in arbitrary diffeomorphism-invariant theories (Wald 1993; Iyer and Wald 1994; Jacobson

et al 1994). This seminal work has revealed, among other properties of black holes, that

the area-entropy relation will only be modified in cases when gravity is supplemented with

nonminimally coupled matter or higher-curvature interactions. This is a consequence of

the fact that such terms modify the gravitational surface term in the symplectic structure.

Such terms naturally arise in the effective actions of superstrings and supergravity. For now

Appendix A. Black Hole Mechanics: An Overview 79

we will restrict our review to D-dimensional black holes of EM theory.

Let us first state some facts about Killing horizons. We shall consider the black hole

region

(A.3)

of a spacetime manifold M; i.e. a region of spacetime that excludes all events that belong

to the causal past of future null infinity. The event horizon is the boundary 8B of the region

B of spacetime. If the black hole is stationary, then the event horizon coincides with the

Killing horizon - a hypersurface at which the vector

L(D-1)/ 2J

'a = ta + I: n~m~ (A.4)

is null. Here, ta is a timelike Killing vector, m~ are l ( D - 1) /2 J rotational spacelike Killing

vectors and the coefficients n~ are the angular velocities of the black hole. The vector (a

is tangent to the null generators of the Killing horizon and therefore satisfies the geodesic

equation

(A.5)

This defines the surface gravity "' of the black hole. Equivalent definitions are given by

2 ; ( ; ;b) d 2 _ 1, ;a;b "'"a= - .,b., ;a an /'i, - - 2'>a;b'> 0 (A.6)

For the Schwarzschild solution (1.2) it can be verified by direct computation that "'

1/(4M0 ).

We now state the four laws of black-hole mechanics:

• Zeroth law. The surface gravity "' is constant over the entire event horizon.

Appendix A. Black Hole Mechanics: An Overview 80

• First law. For a stationary black hole with energy <!:, surface area .91, electric charge

.Q and angular momenta J~, the change in mass during a quasi-static process is given

by

L(D-1)/ 2J

8<!: = _!:_8.Jd + ~8.o + 2::: n~8J~ , "'D ~=1

(A.7)

where~= -Aa(alr=r+ is the electric potential at the horizon (r = r +) and Aa is the

vector potential.

• Second law. The surface area .Jd can never decrease in a physical process if the stress-

energy tensor Tab satisfies the dominant energy condition Tab(a(b 2: 0.

• Third law. The surface gravity cannot be reduced to zero by any physical process in

a finite period of time.

The laws of black-hole mechanics are very similar to the laws of thermodynamics. If

one makes the identification %' = <!:, then the first law of thermodynamics (A.2) and the

first law of black-hole mechanics (A.7) require that Y ex .9/kB/l~ (with physical constants

restored for the moment) and "' ex T. However, the latter does not seem to make sense

from a classical point of view, because a black hole presumably has zero temperature which

would imply that the entropy is infinite. Is the similarity just a mathematical coincidence,

or is nature telling us something deep about the interplay between gravitational phenomena

and quantum mechanics? Indeed, the identification Y ex .sdkB/l~ requires the presence of

c and li in order for the entropy to be dimensionless. It was Bekenstein (Bekenstein 1973;

Bekenstein 1974) who first recognized the physical significance of this similarity. Bekenstein

also recognized that the irreversable process of dropping matter into a black hole leads to

Appendix A. Black Hole Mechanics: An Overview 81

the genemlized second law of thermodynamics:

8.9'universe + 8.9'BH 2: 0 · (A.8)

Using a semiclassical approach, Hawking (1975) then fixed the free parameter to 1/4 and

the temperature toT = r;,j27r. Thus a black hole is not eternal, but rather has a thermody­

namical temperature and radiates. (It should be noted that the final state of this process is

unknown, because the temperature goes to infinity as the surface area decreases. One of the

many goals of all theories of quantum gravity is to give a detailed first-principles description

of the evapouration process and to determine what the final state of a black hole should

be.) Therefore, the laws of black-hole mechanics are the laws of thermodynamics, applied

to an object of special character.

Differential forms

B.l Definitions

Definition B.I: A differential p-form is a totally antisymmentric tensor of type (0, p), i.e.

Denote the vector space of p-forms at a point x by A~ and the collection of p-form fields

by AP. Taking the outer product of a p-form w a1 ···ap and a q-form Jl-b1 ···bq, results in a

tensor of type (0, p + q) , which will not be a (p + q)-form since this tensor is not generally

antisymmetric.

D efinition B.II: The wedge product on an m-dimensional manifold M is a map 1\

A~ x A~ -t A~+q such that the tensor product

is totally antisymmetric.

82

Appendix B. Differential forms 83

Note that this tensor is zero if p + q > m. Thus the wedge product of two one-forms on

JRm2:2 is w 1\ J.£ = - J,£ 1\ w. Now define the vector space of all differential forms at x to be

the direct sum of the A~ such that

The map 1\ : A~ x A~ ---+ A~+q gives Ax the structure of a Grassmann algebra over the

vector space of one-forms.

Definition B.III: The exterior derivative on an m-dimensional manifold M is a map from

the space of p-forms to the space of (p +I)-forms:

together with the following properties:

1. d(w + J.£) = dw + dJ,£ ;

2. d(cw) = cdw ;

3. d(w 1\ ..\) = dw 1\ ,\ + ( -l)Pw 1\ d..\;

4· d(dw) = 0;

Y w , J.£ E AP(M) , ,\ E Aq(M) , and c E JR.

The last property can be easily shown as follows. Consider the p-form w such that

The exterior derivative acting on w is given by

Appendix B. Differential forms 84

and the second exterior derivative is

Since the functions Wa1 ... ap are by definition smooth, the partial derivatives acting on them

commute and the operator 81-Lav is symmetric. The two-form dxi-L 1\ dxv, however, is an-

tisymmetric. Thus we have ddw = 0. This important property is often written simply

In general, if a E AP+1(M) and {3 E AP(M) then: a is called an exact form iff da = 0;

a is called a closed form iff a = d{3. By the last property d2 = 0 of the exterior derivative,

a closed form is automatically an exact form as well. However, the converse is not true, and

the study of this is called DeRham cohomology.

Definition B.IV: The interior product of a p-form w with vector field X on an m-

dimensional manifold M is a map ix : AP(M) ~ AP-l (M) such that

together with the "anti-derivation" property with respect to the wedge product

ix(w 1\ TJ ) = (ixw) 1\ TJ + ( -l)Pw 1\ (ix TJ )

where w E AP(M) and TJ E Aq(M). If w E A0 (M) then ixw = 0.

The interior product is also called the contraction, and is also denoted X ..JW. As an

example, consider w E A2 (M) and Z = X 0 (8jax0) + Y 0 (8jay0

) a vector field on M. The

interior product of w and Z is given by

Appendix B. Differential forms 85

Definition B.V: The Lie derivative of a p-form w with respect to a vector field X is given

by

£xw = X _;dw + d(X _;w ).

Definition B.VI: The Hodge star operator on an m-dimensional manifold M is a linear

map* : AP(M) ---+ Am- P(M) given by

where { ea}~=l is a positively-oriented set of one-forms on some chart of M.

As a simple example, consider a two-form on IR3. Choosing a basis { e1 1\ e2 , e11\ e3 , e2/\

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